KANSAS STATE UNIVERSITY Manhattan, Kansas

Home , Alexander Friedmann, Bharat Ratra, Friedmann equations

CONSTRAINING COMPETING MODELS OF DARK ENERGY WITH

COSMOLOGICAL OBSERVATIONS

ANATOLY PAVLOV

B.S., Saint Petersburg State Polytechnic University, Russia 2005

M.S., Saint Petersburg State Polytechnic University, Russia 2007

AN ABSTRACT OF A DISSERTATION

submitted in partial fulﬁllment of the requirements for the degree

DOCTOR OF PHILOSOPHY

Department of Physics College of Arts and Sciences

KANSAS STATE UNIVERSITY Manhattan, Kansas

2015 Abstract

The last decade of the 20th century was marked by the discovery of the accelerated expansion of the universe. This discovery puzzles physicists and has yet to be fully understood. It contradicts the conventional theory of gravity, i.e. Einstein’s General Relativity (GR). According to GR, a universe filled with dark matter and ordinary matter, i.e. baryons, leptons, and photons, can only expand with deceleration. Two approaches have been developed to study this phenomenon. One attempt is to assume that GR might not be the correct description of gravity, hence a modified theory of gravity has to be developed to account for the observed acceleration of the universe’s expansion. This approach is known as the ”Modified Gravity Theory”. The other way is to assume that the energy budget of the universe has one more component which causes expansion of space with acceleration on large scales. Dark Energy (DE) was introduced as a hypothetical type of energy homogeneously filling the entire universe and very weakly or not at all interacting with ordinary and dark matter. Observational data suggest that if DE is assumed then its contribution to the energy budget of the universe at the current epoch should be about 70% of the total energy density of the universe. In the standard cosmological model a DE term is introduced into the Einstein GR equations through the cosmological constant, a constant in time and space, and proportional to the metric tensor gµν. While this model so far fits most available observational data, it has some significant conceptual shortcomings. Hence there are a number of alternative cosmological models of DE in which the dark energy density is allowed to vary in time and space. CONSTRAINING COMPETING MODELS OF DARK ENERGY WITH

COSMOLOGICAL OBSERVATIONS

ANATOLY PAVLOV

B.S., Saint Petersburg State Polytechnic University, Russia 2005

M.S., Saint Petersburg State Polytechnic University, Russia 2007

A DISSERTATION

submitted in partial fulﬁllment of the requirements for the degree

DOCTOR OF PHILOSOPHY

Department of Physics College of Arts and Sciences

KANSAS STATE UNIVERSITY Manhattan, Kansas

2015

Approved by:

Major Professor Bharat Ratra Copyright

ANATOLY PAVLOV

2015 Abstract

List of Figures ix

List of Tables xiv

Acknowledgements xiv

1 Introduction1 1.1 Theoretical Cosmology ...... 1 1.1.1 Short overview of General Relativity ...... 1 1.1.2 Friedmann Equations ...... 4 1.1.3 Solutions of Friedmann Equations ...... 8 1.1.4 Cosmological Constant ...... 13 1.1.5 Standard Cosmological Model ...... 16 1.2 Observational Cosmology ...... 17 1.2.1 Redshift ...... 17 1.2.2 Hubble parameter ...... 18 1.2.3 Luminosity Distance ...... 19

2 Forecasting cosmological parameter constraints from near-future space- based galaxy surveys 23 2.1 Introduction ...... 23 2.2 Measured power spectrum of galaxies ...... 28 2.3 Cosmological models ...... 32

vi 2.3.1 ΛCDM, XCDM and ωCDM parameterizations ...... 32 2.3.2 φCDM model ...... 33 2.4 Fisher matrix formalism ...... 34 2.5 Results ...... 36 2.5.1 Constraints on growth rate ...... 36 2.5.2 Constraints on dark energy model parameters ...... 37

3 Nonﬂat time-variable dark energy cosmology 47 3.1 Introduction ...... 47 3.2 The Model ...... 49 3.2.1 Solution for the curvature-dominated epoch ...... 51 3.3 Some observational predictions ...... 54

3.3.1 The time parameter H0t0 ...... 55 3.3.2 The distance modulus diﬀerence ∆m(z)...... 57 3.3.3 Number counts ...... 59 3.3.4 The growth of structure ...... 59

4 Cosmological constraints from large-scale structure growth rate measurements 63 4.1 Introduction ...... 63 4.2 Data and analysis ...... 65 4.2.1 Growth rate of LSS ...... 65 4.2.2 SNIa distance modulus ...... 68 4.2.3 Hubble parameter ...... 68 4.2.4 Computation of joint χ2(p)...... 69 4.3 Results and discussion ...... 70

vii 5 Conclusion 74

Bibliography 85

A Appendix for Chapter 2 86

B Appendix for Chapter 3 90

viii List of Figures

1.1 Light cone diagram...... 3 1.2 Cosmic Microwave Background radiation map of the last scattering epoch. Data of 2012 WMAP mission, Source: http://map.gsfc.nasa.gov/media/060913/index.html5 1.3 Two dimensional analogies of hyperbolic, ﬂat, and spherical manifolds. . . .7 1.4 Solutions of Friedmann equations for a single component universes with ﬂat, closed, and open geometries. The time and the scale factor in arbitrary units. 12 1.5 Numerical solution of Eq. (1.30) for standard cosmology. Time and the scale factor presented in arbitrary units, where a = 1 corresponds to the current epoch...... 15

1.6 The plot of log10(ρi/ρcr) vs the log of the scale factor log10(a) for each component. One can see that the early universe was dominated by radiation, however since energy density of radiation dissipates as a−4 while energy density of ordinary matter decreases as a−3, the radiation dominated epoch switches to matter dominated. Eventually, the universe becomes dominated by the cosmological constant, since its energy density does not evolve in time. . . . 16 1.7 Hubble diagram for the Union2.1 compilation. The solid line represents the best-ﬁt cosmology for a ﬂat ΛCDM universe for supernovae alone. Source: Fig.(4) of [12]...... 22

ix 2.1 Predicted relative error on the measurements of growth rate as a function of redshift z in redshift bins of ∆z = 0.1 for different models of dark energy. The upper solid black line shows predictions for the case when no assumption is made about the nature of dark energy...... 40 2.2 Predicted one standard deviation confidence level contour constraints on the current renormalized Hubble constant h and the parameter γ that describes deviations from general relativity for different dark energy models...... 41 2.3 Upper panel shows one standard deviation confidence level contours con-

straints on parameters ωa and ω0 of the ωCDM parametrization, while lower

panel shows these for parameters Ωk and Ωm...... 42 2.4 One standard deviation conﬁdence level contour constraints on parameters of the XCDM parametrization...... 43 2.5 One standard deviation conﬁdence level contour constraints on parameters

α and Ωm of the φCDM model. Lower panel shows a magniﬁcation of the tightest two contours in the lower left corner of the upper panel...... 44 2.6 One standard deviation conﬁdence level contour constraints on parameters

Ωk and Ωm of the φCDM model. The lower panel shows a magniﬁcation of the two tightest contours in the center of the upper panel...... 45 2.7 One standard deviation conﬁdence level contour constraints on parameters

Ωk and Ωm of the ΛCDM model...... 46

x 3.1 Contours of ﬁxed time parameter H0t0, as a function of the present value

of the nonrelativistic matter density parameter Ωm0 and scalar ﬁeld potential power-law index α, at various values of the current value of the space-

curvature density parameter Ωk0 (as listed in the inset legend boxes). The

upper panel shows a larger part of (Ωm0, α) space for a larger range of Ωk0

values [for H0t0 = 0.7, 0.75, 0.8, 0.85, 0.95, 1.05 and 1.15, from right to left], while the lower panel focuses on a smaller range of the three parameters [for

H0t0 from 0.8 to 1.15 in steps of 0.05, from right to left]...... 56 3.2 Contours of ﬁxed bolometric distance modulus relative to the Einstein–de Sitter model, ∆m(z = 1.5), as a function of the matter density parameter

Ωm0 and scalar ﬁeld potential power-law index α, and various values of the

space curvature density parameter Ωk0 (as listed in the inset legend boxes).

The upper panel shows a larger part of (Ωm0, α) space for a larger range of

Ωk0 values [for ∆m(z = 1.5) = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.8 from right to left], while the lower panel focuses on a smaller range of the three parameters [for ∆m(z = 1.5) from 0.3 to 0.8 in steps of 0.1, from right to left]. In the

upper panel there is no Ωk0 = 0.2 contour for ∆m(z = 1.5) = 0.1 since in this case the model is too open for such a small distance modulus diﬀerence. 58

3.3 Contours of ﬁxed A(z = 0.7) as a function of Ωm0 and α at various values

of Ωk0 (as listed in the inset legend boxes). The upper panel shows a larger part of the parameter space for A(z = 0.7) = 0.25, 0.3, 0.35 and 0.45 from right to left. The lower panel shows a smaller range of the three parameters for A(z = 0.7) = 0.3, 0.35, 0.4 and 0.45 from right to left...... 60

xi 3.4 Contours of the factor by which the growth of ordinary matter perturbations

falls below that of the Einstein–de Sitter model, ∆(Ωm0, Ωk0, α), as a function

of the matter density parameter Ωm0 and scalar ﬁeld potential power-law

index α, and various values of the space curvature density parameter Ωk0 (as

listed in the inset legend boxes). The upper panel shows a larger part of (Ωm0,

α) space for a larger range of Ωk0 values [for ∆(Ωm0, Ωk0, α) = 0.3, 0.5, 0.7 and 0.9 from left to right], while the right panel focuses on a smaller range of the

three parameters [for ∆(Ωm0, Ωk0, α) from 0.2 to 0.8 in steps of 0.1, from left

to right]. In the lower panel there is no Ωk0 = 0.2 contour for ∆ = 0.9 since in this case the model is too open to allow such a large growth factor. . . . . 61

4.1 1, 2, and 3σ constraint contours for the ΛCDM model from: growth factor

measurements [blue dashed lines with blue ﬁlled circle at best-ﬁt (Ωm,ΩΛ) =

2 (0.41, 0.87) with χmin/dof = 7.65/12]; SNIa+H(z) apparent magnitude data

[red dot-dashed lines with red ﬁlled circle at best-ﬁt (Ωm,ΩΛ) = (0.23, 0.59)

2 with χmin/dof = 562/598]; and a combination of all data sets [black solid

2 lines and black filled circle at best-fit (Ωm,ΩΛ) = (0.28, 0.69) with χmin/dof = 571/612]. The dashed straight line corresponds to spatially-flat models, the dotted line demarcates zero acceleration models, and the area in the upper left-hand corner is the region for which there is no big bang...... 71

xii 4.2 1, 2, and 3σ constraint contours for the XCDM model from: growth factor

measurements [blue dashed lines with blue ﬁlled circle at best-ﬁt (Ωm, wX ) =

2 (0.36, -1.34) with χmin/dof = 7.70/12]; SNIa+H(z) apparent magnitude data

[red dot-dashed lines with red ﬁlled circle at best-ﬁt (Ωm, wX ) = (0.27, -0.90)

2 with χmin/dof = 562/598]; and a combination of all data sets [black solid

2 lines and black filled circle at best-fit (Ωm, wX ) = (0.28, -0.94) with χmin/dof = 571/612]. The dashed straight line corresponds to spatially-flat ΛCDM models and the dotted curved line demarcates zero acceleration models. . . 72 4.3 1, 2, and 3σ constraint contours for the φCDM model from: growth factor

measurements [blue dashed lines with blue ﬁlled circle at best-ﬁt (Ωm, α) =

2 (0.28, 0.052) with χmin/dof = 8.62/12]; SNIa+H(z) apparent magnitude data

[red dot-dashed lines with red ﬁlled circle at best-ﬁt (Ωm, α) = (0.26, 0.302)

2 with χmin/dof = 562/598]; and a combination of all data sets [black solid

2 lines and black filled circle at best-fit (Ωm, α) = (0.27, 0.300) with χmin/dof = 570/612]. The dotted curved line demarcates zero acceleration models and the horizontal α = 0 axis corresponds to spatially-flat ΛCDM models. . . . 73

xiii List of Tables

2.1 Values of the parameters of the fiducial ΛCDM model and the survey. . . . . 39 2.2 Predicted deviations of parameter γ from its fiducial value, at one standard deviation confidence level, for different assumptions about dark energy. . . . 39

4.1 Growth parameter measurements and 1σ uncertainties. Reference number shown in the last column: 1. [133], 2. [134], 3. [135], 4. [64], 5. [65], 6. [136], 7. [66], 8. [63], 9. [137]...... 66

A.1 Values of the kmax, bias b(z) from [87], and the number densities n(z) taken from [88]...... 89

xiv Acknowledgments

I greatly appreciate the help and support of so many people among my colleagues and friends while writing this dissertation. I would like to thank specially my research adviser Professor Bharat Vishnu Ratra who was advising me through my research and studies at Kansas State University. Thank you for your help and patience in educating me in the most complicated areas of physical cosmology and data analysis. I am grateful to Lado Samushia for his priceless help while I was working on my research as well as on projects we have worked in collaboration. Lado has helped me a lot in the most complicated technical caveats of the research activities of our ﬁeld. I am indebted to Mikhail Makouski and Data Mania for their help with programming which was the main technical skill required to successfully conduct my research. I am very thankful to Professor Tim Bolton for thoughtful conversations and tricky questions during High Energy Physics seminars, which helped me to reveal many details in areas of experimental as well as theoretical physics. I am also grateful to Professor Tim Bolton for his guidance while I was teaching for Engineering Physics I and II studios. Special thanks to Professors Yurii Maravin and Tina Kanhiashvili; its mainly due to their suggestion and help that I joined Kansas State University as a graduate student. I am endlessly thankful to Professor Larry Weaver for discussions we had about various subjects of theoretical physics. Thank you very much for listening to my ideas and giving me suggestions on how to bring them to life by using mathematics to develop a framework of a physical model. I also would like to thank Professor Sanjoy Das for his help in understanding of how modern Machine Learning techniques work and how can it be applied to problems of data analysis in physical cosmology. Thank you very much.

xv Chapter 1

Introduction

1.1 Theoretical Cosmology

Cosmology is the study of the universe as a whole. Since ancient times people across the globe have tried to understand the origins of the universe and predict its future. However, it was not until the beginning of the 20th century when cosmology emerged as a quantitative scientiﬁc ﬁeld rather than a philosophical attempt to make sense of the universe in a mytho- logical manner. The theoretical basis of cosmology lies mainly on Einstein’s description of gravity, i.e. General Relativity, a theory in which gravity is described as the geometry of space-time, rather than a force, where the geometry is determined by the matter content and its distribution. On the other hand, the geometry of space-time governs the motion of matter.

1.1.1 Short overview of General Relativity

Diﬀerential geometry provides a mathematical framework for general relativity. The diﬀer- ential of distance between two near by points in space-time, xµ and xµ + dxµ, is given by

1 the line element which is deﬁned via the following quadratic form

2 µ ν ds = gµνdx dx (1.1)

µ where gµν is the metric tensor and x are coordinates. In general relativity coordinates describe 4-dimensional space-time with one-time dimension and three-dimensions of space. The coordinate xµ = (x0, x1, x2, x3) where x0 = ct with c being the speed of light and

1 2 3 t is time, while x , x , x label spatial part of the space-time. The metric tensor gµν is a fundamental object in general relativity. It may loosely be thought of as gravitational potentials in Newtonian gravitation terms. It has a total of 16 components but due to

2 its symmetry gµν = gνµ there are only 10 independent components. The line element ds describes the casual structure of space-time. If ds2 > 0, the interval is time-like and the line element is the proper time interval between the events in space-time traversed by massive particles. When ds2 = 0, the interval is light-like and the line element is the proper time interval between events in space-time which can only be traversed by mass-less particles. In the case when ds2 < 0, the interval is space-like and the absolute value of the line element describes the proper length between events in space-time that cannot be in casual connection with one another. The structure of space-time can be represented through the light-cone diagram which is shown on Fig. (1.1). We now discuss how the geometry of space-time interacts with its matter content and how geometry governs dynamics of motion of matter. The mechanism of interaction between geometry and matter is what general relativity is all about, and the Einstein equations give a quantitative description of this interaction. The Einstein equations are

1 8πG R − Rg = T , (1.2) µν 2 µν c4 µν

µν where G is Newton’s gravitational constant, R = g Rµν the scalar curvature, and Rµν is

2 ct

ds2>0 Future

ds2<0

0 2 = Past ds

Figure 1.1: Light cone diagram. the Ricci tensor ∂Γλ ∂Γλ R = µν − λµ + Γλ Γκ − Γλ Γκ . (1.3) µν ∂xλ ∂xν λκ µν µκ λν

λ Here Γµν are the Christoﬀel symbols deﬁned as

1 ∂g ∂g ∂g Γλ = gλκ κµ + κν − µν . (1.4) µν 2 ∂xν ∂xµ ∂xκ

The Ricci tensor is symmetric Rµν = Rνµ as can be seen from the deﬁnition above, thus

3 there are only 10 independent components. The term on the right hand side of Eq. (1.2) is the energy-momentum tensor Tµν that describes matter. Einstein’s equations of general relativity form a system of 10 non-linear equations of second order with respect to metric tensor gµν. They reduce to Newton’s gravity law in the weak gravity limit. The energy- momentum tensor Tµν must satisfy a conservation equation of the following form in general

µ relativity T ν;µ = 0 and it expresses energy and momentum conservation laws. Semicolon in the equation of energy-momentum conservation refers to co-variant differentiation which is the appropriate way of computing derivatives in differential geometry. The conservation equation played a major role when Einstein was trying to guess the right form of the equations that describe how matter curves space. The very first equation which Einstein wrote was simply Rµν ∼ Tµν but he quickly discovered that the Ricci tensor alone does not satisfy the conservation equation. However, the tensor 2Rµν − Rgµν does satisfy the conservation condition and is what Einstein stated as the equations of general relativity. The tensor on the left hand side of Eqs. (1.2) Rµν − Rgµν/2 is called the Einstein tensor and labeled Gµν in the literature.

1.1.2 Friedmann Equations

Observational data of Cosmic Microwave Background (CMB) radiation anisotropy, see Fig. (1.2), and other measurements [1] indicate that the universe is homogeneous on scales beyond 100 Mpc while the observable part of the universe is about 4000 Mpc, which makes it reasonable to chose spatially homogeneous and isotropic metrics to describe the space- time of the universe. The appropriate form of metrics for the universe was introduced by Alexander Friedmann in 1922 where he allowed them to be time-variable. Friedmann showed that the universe would be dynamic and would expand six years before the expansion was observed by Edwin Hubble in 1928. The line element in Friedmann’s metric is

2 2 2 i j ds = −dt + a (t)γijdx dx (1.5)

4 Figure 1.2: Cosmic Microwave Background radiation map of the last scattering epoch. Data of 2012 WMAP mission, Source: http://map.gsfc.nasa.gov/media/060913/index.html

where we chose units in which c = 1 and metric tensor sign convention (− + ++). The indexes are labeled with Greek letters when they run in 4-dimensional space-time, i.e. from 0 to 3, and when they run in the 3-dimensional spatial part of space-time labeling is done with Latin letters that go from 1 through 3. The function a(t) is called the scale factor and it describes how the distance between two points in space evolves with time. The term

2 a (t)γij = gij is the spatial part of the metric tensor. Because of the assumption of isotropy and homogeneity, adopted based on observational evidence, the metric tensor has to be diagonal, hence, gij is also diagonal. The Friedmann equations are a special case of Einstein’s equations of general relativity Eq. (1.2) for metrics of Eq. (1.5). These reduce the Einsteinian system of 10 equations to a system of two equations for one unknown function, the scale factor a(t),

a˙ 2 k 8πG + = ρ, (1.6) a a2 3 a¨ 4πG = − (ρ + 3p) . (1.7) a 3

Here ρ is the matter density and p is pressure. The left hand side of Eqs. (1.6)-(1.7) are

5 derived from the left hand side of Einstein’s equations (1.2) which describe the geometry of space-time, while the right hand side of equations (1.6)-(1.7) are components of the energy-momentum tensor Tµν. In particular Eq. (1.6) corresponds to equality between the time components of Einstein and energy-momentum tensors G00 = 8πGT00 and Eq. (1.7) corresponds to equality between diagonal elements of the spatial part of Einstein and energy- momentum tensors Gii = 8πGTii and is derived with the use of Eq. (1.6). For more detailed derivation see paragraph 2.1.3 of [2]. Eqs. (1.6)-(1.7) are written for a universe that has only one form of matter that ﬁlls it. If there are multiple constituents present, then Eqs. (1.6)-(1.7) have to be generalized by replacing mass density ρ and pressure p with sums over all types of matter i.e. Σρi → ρ and

Σpi → p. In order to solve the Friedmannian equations one has to know how density and pressure depend on time. In some cases pressure and density can be related by an equation of state whose exact form depends on the particular form of fluid. For example, in the case of barotropic fluids pressure is a function of mass density only i.e., p = p(ρ), hence one needs to know only how mass density varies with time. There are three possible spatial geometries that meet requirements of homogeneity and isotropy of the universe. These are: flat, spherical, and hyperbolic geometries. Flat or non-curve space, is a space which obeys all properties of Euclidean geometry, and in general relativity it can be dynamic. Spherical and hyperbolic spaces do not obey the axioms of Euclidean geometry; these types of spaces are curved. Euclidean relationships of elemental figures and objects of geometry are not satisfied in curved spaces. For example, the sum of all angels of a triangle need not equal π and the ratio of circumference of a circle to its radius r need not equal 2πr. In Fig. (1.3) two dimensional analogies of hyperbolic, flat, and spherical spaces are shown. Mathematical description of spaces presented in Fig. (1.3) are given by the following

6 Figure 1.3: Two dimensional analogies of hyperbolic, ﬂat, and spherical manifolds. metrics, written in Cartesian coordinates

a2(t) g = δ , (1.8) ij 1 − kr2 ij

2 i j where r = δijx x is square of distance between two points in space, δij is the Kronecker delta symbol and k is the curvature parameter which takes values k = 0 for flat space, k = 1 for spherically curved space, and k = −1 in case of hyperbolic geometry. The normalization of the absolute value of curvature parameter |k| = 1 is chosen for convenience. Geometry of space determines significant two properties of the universe. One property is whether our universe is finite or infinite and the other characteristic is if the universe is eternal or has a finite life span from the Big Bang to what is proposed as the opposite of the Big Bang, the Big Crunch. The universe is finite in case of spherical geometry of space as it is shown in Fig. (1.3). Such a universe has a limited volume and because of that is called a “closed” universe. A closed universe, filled with ordinary and dark matter alone, has a finite life span, but if it contains dark energy there is a set of solutions which allow for an eternal universe with and without the Big Bang. A non-eternal closed universe expands from the Big Bang, reaches its maximum volume, and then contracts until it reaches the Big Crunch.

7 In the other two cases of geometry i.e. flat or hyperbolic the universe is infinite and will expand forever after the Big Bang. These two scenarios of infinite and eternal universes are also called “open” universes. In order to determined which universe we live in, its curvature parameter has to be measured.

1.1.3 Solutions of Friedmann Equations

Flat Matter and Radiation Dominated Universes, k = 0: Since the second Friend- mann equation (1.7) does not contain curvature parameter explicitly, reviewing different scenarios of the universe’s dynamics with respect to its curvature will involve only the first Friedmann equation (1.6), which for the flat universe is

2 a˙ m 8πG = ρm (1.9) am 3 here the subscript m denotes the universe is filled with non-relativistic dust-like fluid and called a matter dominated universe. It is important to emphasize that the Eq. (1.9) expresses a condition for matter density to yield flat universe and for this reason the density defined as 3 a˙ 2 ρ = (1.10) cr 8πG a is called critical density of the universe. Substituting the equation of state pm(ρm) = 0 of non-relativistic matter in the second Friedman equation (1.7) and rewriting it with the use of Eq. (1.9) yields the conservation equation

a˙ m ρ˙m + 3ρm = 0, (1.11) am

3 from which one recovers am(t)ρm(t) = const. In what follow, I will use the following conditions: the Big Bang time tBB = 0, the present time will be labeled t0 and it also will be the age of the universe, the scale factor for the Big Bang and current epoch satisﬁes

8 a(tBB) = aBB = 0, a(t0) = a0 = 1, and ρi(t0) = ρi0 is the present value of matter density of the type “i”. With the conditions adopted above the solution of the Eq. (1.9) is

1/3 2/3 am(t) = (6πGρm0) t . (1.12)

This solution is known as the Einstein-De Sitter model. In case when the universe dominated by relativistic type of ﬂuid, i.e. particles that move with speeds close to the speed of light (neutrinos for example) or mass-less particles

(photons), the equation of state is the same as for radiation pr = ρr/3, where subscript r denotes radiation and thus this type of universe is called radiation dominated. Substitution of this equation of state in the second Friedmann Eq. (1.7) results in the conservation relation

4 ar(t)ρr(t) = const, which provides the following solution of Eq. (1.9),

2 1/4 a (t) = πGρ t1/2. (1.13) r 3 r0

Closed Matter and Radiation Dominated Universes, k = 1: Now, let us derive solutions for the previous two matter constituents but in presence of positive curvature parameter k = 1 which corresponds to closed spherically curved universe. First I will solve Friedmann equations for non-relativistic matter dominated case i.e.

2 a˙ m 1 8πG + 2 = ρm, (1.14) am am 3

−3 where ρm ∼ a . The solution in this case cannot be obtained as an explicit function of time however, parameterization through conformal time, deﬁned as dη = dt/a(t), yields the

9 cycloidal solution

q0 am(η) = [1 − cos(η)] , (1.15) 2q0 − 1

q t(η) = 0 [η − sin(η)] . (1.16) 2q0 − 1

Here q0 is the value of the deceleration parameter at the current epoch q0 = q(t0) where the deceleration parameter is defined as aa¨ q = − , (1.17) a˙ 2 and its physical meaning is that it gives a quantitative measure of the deceleration rate of the universe’s expansion. It can be rewritten with the use of Friedmann equations (1.6)-(1.7) as ρ q = (1.18) 2ρcr from which one can notice that the deceleration parameter can also be thought of as a measure of how much the universe deviates from being flat in case when it has curvature and deceleration parameter q 6= 1/2. In order to quantify more strictly how the deceleration parameter q depends upon the curvature parameter k one has to substitute the expression for the matter density ρ from the first Friedmann equation (1.6) into the Eq. (1.18) from which one can retrieve the following

1 k = 1 then q > (1.19) 2 1 k = −1 then q < . (1.20) 2

Relations (1.19)-(1.20) show that when matter density is higher than critical density the universe is closed while when the matter density is less than critical density the universe is open. If one introduces the dimensionless density as the ratio Ω = ρ/ρcr then the deceleration

10 parameter can be simply written as q = Ω/2. In this particular example of a matter dominated universe one has ρm, hence q = Ωm/2, where Ωm = ρm/ρcr. In case of a radiation dominated universe one has

2 a˙ r 1 8πG + 2 = ρr, (1.21) ar ar 3

−4 where ρr ∼ a . The solution is

r 2q0 ar(η) = sin(η), (1.22) 2q0 − 1

r 2q t(η) = 0 [1 − cos(η)] , (1.23) 2q0 − 1

where q0 = Ωr0/2. Open Matter and Radiation Dominated Universes, k = −1: The Friedmann equation for an open universe is

2 a˙ i 1 8πG − 2 = ρi, (1.24) ai ai 3 where i takes the values m for a matter dominated universe and the value r for a radiation dominated universe. The solution for the matter dominated universe is

q0 am(η) = [cosh(η) − 1] , (1.25) 1 − 2q0

q t(η) = 0 [sinh(η) − η] . (1.26) 1 − 2q0

11 And for radiation dominated case the solution is

r 2q0 ar(η) = sinh(η), (1.27) 1 − 2q0

r 2q t(η) = 0 [cosh(η) − 1] . (1.28) 1 − 2q0

The solutions we derived are shown below in Fig. (1.4).

40 Flat−Matter Flat−Radiation Closed−Matter 35 Closed−Radiation Open−Matter Open−Radiation 30

25 a(t) 20

0 0 5 10 15 20 25 30 35 40 t

Figure 1.4: Solutions of Friedmann equations for a single component universes with ﬂat, closed, and open geometries. The time and the scale factor in arbitrary units.

In the above we reviewed examples of ideal universes which contain one type of matter i.e. non-relativistic dust or radiation. The real universe consists of both types of matter and possibly dark energy. In the section below I investigate more realistic solutions of the

12 Friedmann equations for a universe with multiple types of matter content.

1.1.4 Cosmological Constant

The cosmological constant was ﬁrst introduced by Albert Einstein when he was attempting to apply his theory of general relativity to the entire universe. It was commonly assumed in the early 20th century that the universe was static. However, since gravity is an attractive force with bottomless potential to ﬁnd a static solution Einstein introduced the cosmological constant labeled by Greek letter Λ, in his equations

1 R − Rg + Λg = 8πGT . (1.29) µν 2 µν µν µν

The effect of the cosmological constant is to causes space to expand as if there were a repulsive gravitational force. Einstein’s idea was to use the effect of repulsion from the cosmological constant to cancel out ordinary matter’s attraction and so have a static solution for the universe. However, it was shown that even with a cosmological constant a static solution is not stable with respect to small perturbations in matter density and eventually the idea of a static universe was abandoned after Hubble’s discovery of the expansion of the universe. Einstein called his invention of a cosmological constant as his biggest blunder. However, since Einstein’s times scientists keep coming back to the idea of a cosmological constant over and over again because of theoretical physics considerations such as the quantum vacuum and cosmic inflation. However, it was not until the discovery of the accelerated expansion of the universe in the late 1990’s which provided the first observational evidence for a cosmological constant. The way Einstein thought of the cosmological constant was that it was a fundamental constant of nature and was part of the gravity law, thus Einstein was the first who developed a “Modified Gravity Theory” with Λ placed on the left hand side of Eq. (1.29), i.e. geo- metrical part of the Einstein equations. The modified gravity as an alternative explanation

13 of the accelerated cosmological expansion is highly discussed topic in modern cosmology, however I do not address this approach in a great details. For recent reviews of modified gravity see [3] and [4]. When cosmological constant is written on the right hand side of Einstein’s equations it introduces a type of energy that fills the universe homogeneously and causes repulsion on the scales determined by the cosmological constant. This view gives the first Friedmann equation as a˙ 2 8πG ρ ρ ρ = r0 + m0 + k0 + ρ , (1.30) a 3 a4 a3 a2 Λ where ρk0 = −3k/8πG is the present value of “curvature density” chosen to represent curvature as if it were a type of matter and the meaning of this representation is that it shows amount of matter density deficit in the universe to make it flat when it has hyperbolic geometry while in case of spherically curved universe ρk represents how much extra matter density in the universe over critical density. Here ρΛ is the energy density associated with the cosmological constant Λ ρ = , (1.31) Λ 8πG which is constant in time with equation of state

pΛ = −ρΛ. (1.32)

Energy density associated with a cosmological constant is called Dark Energy, where “dark” refers to the fact that it does not shine like ordinary Byronic matter i.e. protons, nuclei, ions, atoms, and electrons. The Eq. (1.30) does not have analytic solution in general case. Numerical solutions for standard cosmological model are presented in Fig. (1.5). It is worth to emphasize how diﬀerent constituencies that ﬁll the universe evolve with time after the

Big Bang. For that purpose I plot log10(ρi/ρcr) vs the log of the scale factor log10(a) for each component in a single graph shown in Fig. (1.6). Motivation for a cosmological constant as a type of energy comes from quantum ﬁeld

14 2.5

1.5 a(t)

0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

Figure 1.5: Numerical solution of Eq. (1.30) for standard cosmology. Time and the scale factor presented in arbitrary units, where a = 1 corresponds to the current epoch. theory. In quantum mechanics absolute vacuum as empty space does not exist. Quantum mechanics predicts ﬂuctuations of energy in otherwise empty space whose average value yields total non-zero energy density always present in space called the Zero-Point Energy Density whose value for a scalar ﬁeld with mass m can be expressed as [e.g.,5]

m4 m2 ρ = ln , (1.33) vac 64π2 µ2 where µ is renormalization scale. The equation of state of the vacuum energy is the same as the equation of state for dark energy, Eq. (1.32), i.e. pvac = −ρvac, which is what supports the assumption of the quantum vacuum as a possible candidate for dark energy.

However, computation of the vacuum energy density predicts it to be in the range ρvac '

8 72 4 −48 4 10 −10 GeV while ρΛ ' 10 GeV is observed which deviates from the quantum vacuum prediction by 120 orders of magnitude! This inconsistency is known as the quantum vacuum

15 20 Radiation Matter Curvature 15 Cosm.Const.

10 ) cr ρ / i ρ

( 5 10 log

−5

−10 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 log (a) 10

Figure 1.6: The plot of log10(ρi/ρcr) vs the log of the scale factor log10(a) for each component. One can see that the early universe was dominated by radiation, however since energy density of radiation dissipates as a−4 while energy density of ordinary matter decreases as a−3, the radiation dominated epoch switches to matter dominated. Eventually, the universe becomes dominated by the cosmological constant, since its energy density does not evolve in time. problem in cosmology.

1.1.5 Standard Cosmological Model

According to general relativity (GR), any form of energy aﬀects space-time dynamics and so cosmological evolution. This fact allows for a very simple phenomenological explanation of the observed accelerated expansion, attributing it to a cosmological constant Λ, homogeneously distributed in space and constant in time. This ΛCDM model [6] is now accepted as the standard cosmological model. At the current epoch Λ dominates the energy budget, with nonrelativistic cold dark matter (CDM) being the next largest contributor, followed by ordinary baryonic matter in third place. A widely discussed generalization of ΛCDM is

16 the φCDM model in which Λ is replaced by a time-varying dark energy density modeled by a self-interacting scalar ﬁeld φ [7]. For recent reviews see [8], [9], [10], and [11]. Observational data available [1] indicate that energy budget of the universe at the current epoch is largely dominated by dark energy contributing about 68.25%, with 26.86% for cold dark matter and 4.89% of baryonic matter. Curvature is constrained to less than 2.00% of the energy budget if the cosmological constant is dark energy. See Table 2 of [1] for the most recent and complete list of the parameters of the standard cosmological model or ΛCDM cosmology, where CDM stands for Cold Dark Matter and Λ refers to dark energy modeled by cosmological constant Λ. It is common in the ﬁeld of physical cosmology to name alternative cosmological models of dark energy in a similar way by replacing only Λ with a symbol that refers to a particular alternative form of dark energy.

1.2 Observational Cosmology

In previous sections I have reviewed theoretical aspects of physical cosmology. In this section I provide a short overview of observational cosmology.

1.2.1 Redshift

The scale factor is a fundamental object in cosmology, although its absolute value does not have any useful physical meaning. What does have physical meaning and can be measured is the ratio of the scale factor taken between diﬀerent epochs. It is a quantitative measurement to how much the universe has expanded between those two epochs. This ratio, as will be shown later, comes into many observable quantities. How can the ratio of scale factors at diﬀerent epochs be measured? The physical meaning of expansion of space is that if one chooses any two arbitrary points in space with initial separation l0 this physical distance will evolve in time approximately proportional to the scale factor i.e. l(t) ∼ l0a(t). In general relativity time-evolving spatial metrics causes a change in the wavelength of electromagnetic

17 radiation which can be expressed as

λemaobs = λobsaem, (1.34)

where λem is the photon’s wavelength when it was emitted by its source at epoch aem and

λobs is its wavelength when it was observed at epoch aobs. This change in wavelength results in a so called, cosmological redshift. Redshift is when the wavelength of electromagnetic radiation increases. The cause of increase of wavelength determines the type of redshift. Thus there is a Doppler redshift, due to motion away from the source of radiation, gravitational redshift, due to radiation propagating away from the source of the gravity field, and cosmological redshift, due to the expansion of space. The word “redshift” originated from the optical effect when the wavelength of radiation increases towards the red part of the spectrum. The definition of redshift is

λ − λ z = obs em , (1.35) λem which, after substituting cosmological wavelength change via scale factor given in Eq. (1.34) and recalling that aobs = 1, yields the equation for cosmological redshift

1 z = − 1. (1.36) a

Redshift is a directly measured quantity determined from analysis of spectral lines emitted by the observed source of radiation.

1.2.2 Hubble parameter

The name “Hubble parameter” originates from the Hubble law discovered by astronomer Edwin Hubble in 1928, which states that the velocity of recession v of a galaxy from an observer anywhere in the universe is proportional to the physical distance l between the

18 galaxy and observer, and the coeﬃcient of proportionality is the Hubble constant H0. Thus, the Hubble law is

v = H0l, (1.37)

˙ where v = l and H0 is the value of the Hubble parameter at the present epoch. Hubble law was ﬁrst derived from the Einstein equations by George Lemaˆıtrein 1927 who independently derived the same equations of motion for the universe as Friedmann did. Recalling from the ˙ previous section l(t) ∼ l0a(t), which after diﬀerentiating yields l ∼ l0a˙ = v and v/l =a/a ˙ , therefore one has for the Hubble parameter the following

a˙ H = . (1.38) a

As we can see the Hubble parameter is not a constant. Now we can rewrite the ﬁrst Friedmann equation in terms of measurable quantities, which is used most of the time for practical purposes in physical cosmology. Below is an example for a universe ﬁlled with radiation, matter, cosmological constant, and non zero curvature

2 2 4 3 2 H (z) = H0 Ωr0(1 + z) + Ωm0(1 + z) + Ωk0(1 + z) + ΩΛ . (1.39)

1.2.3 Luminosity Distance

Distance measurement in astronomy is based on the parallax method and provides direct measurement up to about 100 pc; any distance beyond this limit has to be measured with the use of indirect techniques. Physical cosmology deals with intergalactic objects, thus distances involved are of order 106 pc. A major method widely used in cosmology is based on observations of Supernovae explosions of a speciﬁc type called Ia (reads as “one a”). Supernovae of type Ia are the only to be stars from binary systems where one of the stars is a white dwarf that is orbiting close enough to its partner so that it strips matter from it which accretes back to the white dwarf causing it to increase its mass until it reaches the so

19 called Chandrasekhar limit. At this point the white dwarf’s internal electron pressure unable to resist against gravitational pull and the star collapses on itself triggering thermonuclear reactions in its interior which results in an explosion. Thus allowing for a determination of their absolute luminosity. Supernovae Ia are “standard candles” in determining distances (obs) on intergalactic scales. Measuring flux at a distance rL from a star can be used to figure out that distance via the relationship between absolute luminosity and observed flux

Lem Fobs = 2 , (1.40) (obs) π 2rL

where Fobs is observed ﬂux and Lem emitted absolute luminosity. This is closest method to what could be considered as a direct method of distance measurement. It is independent of the cosmological model, thus it can be used to study diﬀerent cosmologies. Theoretical prediction of distance traveled by photon between two points in space can be derived from the equation of the line element Eq. (1.1), or for our purposes we use the line element written in Friedmann metrics Eq. (1.5). For a photon ds2 = 0, so one concludes that the distance traveled by a photon written in spherical coordinates is

dr dt = a(t)√ , (1.41) 1 − kr2 √ where dr/ 1 − kr2 is the coordinate distance between two neighboring points that does not change as space expands or contracts. The physical distance however, does change as space expands or contracts. It is very easy to see from Eq. (1.41) that in case of ﬂat space the coordinate distance between an object that emitted light at the time tem and an observer receiving the signal at the current epoch t0 is

Z t0 dt r = . (1.42) tem a(t)

20 In the more general case with non-ﬂat spatial geometry one can derive the expression

1 p Z t0 dt r = √ sin −Ωk0 , (1.43) −Ωk0 tem a(t)

here Ωk0 is the present value of the curvature energy density deﬁned as Ωk0 = ρk0/ρcr0. Let us rewrite Eq. (1.43) using redshift as a variable. Expressing dt = da/a˙ and using Eq. (1.39) fora/a ˙ Eq. (1.43) transforms to

1 Z z dz0 √ p r(z) = sin −Ωk0 0 . (1.44) −Ωk0 0 H(z )

This determines the relation between coordinate distance and redshift. However, coordinate distance is not directly observable since observations done via measurements of the flux of radiation coming from distant which lose energy due to redshift, thus the physical distance (obs) rL in Eq. (1.40) is different from coordinate distance by a factor (1+z) which accounts for (obs) the decrease in observable flux due to redshift. The distance rL called physical or photo- metric or luminosity distance and its physical meaning is it gives estimate of how much the flux from a star with redshift z decreased by the time it reached the observer. Therefore the appropriate theoretical physical distance suitable for comparison with observable physical distance is 1 + z Z z dz0 (th) √ p rL (z) = sin −Ωk0 0 . (1.45) −Ωk0 0 H(z ) Distance measurements are expressed using the distance modulus µ = m − M, which is the difference between the apparent magnitude m and the absolute magnitude M of an astronomical object. It is related to the luminous distance rL as

µ = 5log10(rL) − 5. (1.46)

In the Fig. (1.7) the collection of the most recent Supernovae Ia data of Union 2.1 compilation from Ref. [12] are shown.

( :;

(

( ;

(

! &

2 #

'1#

( :;

;

)

( :;

* >

* ¤

(

)

(

)

£ £ ¦

( (

(

¢¡

(

¡ ¡

' '

! !

& &

% %

2 #

" #

¨ ©

-/.

& =

' -

¥ D

B + B

( :;

)

¢¡

)

CDM universe for supernovae alone. Source: Fig.(4) of [12]

(

)

) *

(

)

(

! ¡

(

9 ¡

% 2

¢¡

'/.

-1#

" #$

-/.0

7 Hubble diagram for the Union2.1 compilation. The solid line represents the

+ 7 8 + ,

0 5

;

¢¡

£ ¦

¥ ¤

¤ ¥

¤ ¤

Figure 1.7 best-ﬁt cosmology for a ﬂat Chapter 2

Forecasting cosmological parameter constraints from near-future space-based galaxy surveys

2.1 Introduction

Recent measurements of the apparent magnitude of Type Ia supernovae (SNeIa) continue to indicate, quite convincingly, that the cosmological expansion is currently accelerating [see, e.g., 12–15]. There is good observational evidence that the large-scale radiation and matter distribu- tions are statistically spatially isotropic. The (Copernican) cosmological principle, which is also consistent with current observations, then indicates that the Friedmann-Lemaˆıtre- Robertson-Walker (FLRW) models provide an adequate description of the spatially homogeneous background cosmological model. In the FLRW models, the current accelerating cosmological expansion is a consequence of dark energy, the dominant, by far, term in the current cosmological energy budget. The dark energy density could be constant in time (and hence uniform in space) — Einstein’s

23 cosmological constant Λ [6] — or gradually decreasing in time and thus slowly varying in space [7]. The “standard” model of cosmology is the spatially-flat ΛCDM model in which the cosmological constant contributes around 68.25% of the current energy budget. Non-relativistic cold dark matter (CDM) is the next largest contributor, at around 26.86%, with non- relativistic baryons in third place with about 4.89%. For a review of the standard model see [16] and references therein. Recent measurements of the anisotropies of the cosmic microwave background (CMB) radiation [e.g., 17, 18], in conjunction with significant observational support for a low density of non-relativistic matter [CDM and baryons together, e.g., 19], as well as measurements of the position of the baryon acoustic oscillation (BAO) peak in the matter power spectrum [e.g., 20–23], provide significant observational support to the spatially-flat ΛCDM model. Other data are also not inconsistent with the standard ΛCDM model. These include strong gravitational lensing measurements [e.g., 24–26], measurement of Hubble parameter as a function of redshift [e.g., 27–30], large-scale structure data [e.g., 31–34], and galaxy cluster gas mass fraction measurements [e.g., 35–37]. For recent reviews of the situation see, e.g., [38], [39], and [40]. While the predictions of the ΛCDM model are in reasonable accord with current observations, it is important to bear in mind that dark energy has not been directly detected (and neither has dark matter). Perhaps as a result of this, some feel that it is more reasonable to assume that the left hand side of Einstein’s Eq. (1.2) needs to be modified (instead of pos- tulating a new, dark energy, component of the stress-energy tensor on the right hand side). While such modified gravity models are under active investigation, at present there is no compelling observational reason to prefer any of these over the standard ΛCDM cosmological model. The ΛCDM model assumes that dark energy is a cosmological constant with equation

24 of state

pΛ = −ρΛ, (2.1)

where pΛ and ρΛ are the pressure and energy density of the cosmological constant (ﬂuid). This minimalistic model, despite being in good agreement with most observations available today, has some potential conceptual shortcomings that have prompted research into alternative explanations of the dark energy phenomenon.1 To describe possible time-dependence of the dark energy density, it has become popular to consider a more general equation of state parametrization

pω = ω(z)ρω. (2.2)

Here pω and ρω are the pressure and energy density of the dark energy ﬂuid with redshift z dependent equation of state parameter ω(z). The simplest such parametrization is the XCDM one for which the equation of state parameter is constant and results in accelerated expansion if ω(z) = ωX < −1/3. In this case the dark energy density decreases with time and this allows for the possibility that the fundamental energy density scale for dark energy is set at high energy in the early Universe and the slow decrease of the energy density over the long age of the Universe ensures that the characteristic dark energy density scale now is small (a few meV). This also ensures that the dark energy density remains comparable to the matter energy density over a longer period of time (compared to that for the ΛCDM model).

When ωX = −1 the XCDM parametrization reduces to the consistent (and complete)

ΛCDM model. For any other value of ωX the XCDM parametrization cannot consistently describe spatial inhomogeneities without further assumptions and extension [see, e.g., 43, 44]. Models in which ω(z) varies in time, ωCDM models, are also unable to consistently describe spatial inhomogeneities without further assumptions and extension.

1 Structure formation in the ΛCDM model is governed by the “standard” CDM structure formation model, which might be in some observational diﬃculty [see, e.g., 41, 42].

25 A physically and observationally viable alternative to the ΛCDM model, that consistently describes a slowly decreasing in time dark energy density, is the φCDM model [7, 45]. This model, in which a dark energy scalar field, φ, slowly rolls down its potential, resulting in a slowly decreasing dark energy density, alleviates some of the conceptual problems, mentioned above, associated with the ΛCDM model. The slowly rolling scalar field, at a given instant of time, can be approximated by a dark energy fluid with an appropriately negative equation of state parameter. More specifically, a φCDM model with an inverse-power-law scalar field potential energy density V (φ) ∝ φ−α, α > 0, is a prototypical example that has been extensively studied. This model has a non-linear attractor or “tracker” scalar field solution that forces the initially sub-dominant dark energy density to come to dominate over the matter energy density, thus dominating the energy budget of the current Universe, and so resulting in the current accelerated cosmological expansion. In addition to therefore partially alleviating the “coincidence” problem of the ΛCDM model, the φCDM model generates the current tiny dark energy scale of order an meV, measured by the SNeIa, through decrease, via cosmological expansion over the long age of the Universe, of a much larger energy scale. The α parameter controls the steepness of the scalar field potential, with larger values resulting in a stronger time dependence of the approximate equation of state parameter and α = 0 corresponds to the ΛCDM model limit. α has been constrained using currently available data [see e.g., 46–49, and references therein]. The strongest current limits are that α has to be less than ∼ 0.7 at 2σ confidence [50]. In the φCDM model, or in the XCDM or ωCDM parameterizations, the background evolution of the (spatially homogeneous) Universe differs from that in the ΛCDM case. This affects both the distance-redshift relation as well as the growth rate of large-scale structure. With precise measurements of distance and growth rate over a range of redshifts it will be possible to discriminate between cosmological models.2

2 There are many other models under current discussion, besides the ΛCDM and φCDM models and XCDM and ωCDM parameterizations we consider here for illustrative purposes. For a sample of the available

26 The BAO signature in the observed large-scale structure of the Universe allows for the measurements of radial and angular distances as functions of redshift [see, e.g., 20, 23, 59, 60]. In addition, the redshift-space distortion signal allows for inferences about the strength of gravitational interactions on very large scales [see, e.g., 61–66]. Currently available data sets have been used to measure distances and growth history up to a redshift z ∼ 0.8 and the next generation of planned space-based galaxy redshift surveys of the whole extragalactic sky are expected to extend these measurements to a redshift z ∼ 2. Possible candidates for such surveys include the Euclid satellite mission that has been approved by the European Space Agency [67] and the WFIRST satellite that was ranked high by the recent Decadal Survey [68]. These surveys have been shown to have the potential to measure angular distances, Hubble parameter H(z), and growth rate as functions of redshift to a few percent precision over a wide range of redshifts [69–72].3 As mentioned above, an alternative potential explanation of the observed accelerated expansion of the Universe is to replace general relativity by a modified theory of gravity. For example, in the f(R)-gravity models the Einstein-Hilbert gravitational action is modified to 1 Z √ S = d4x −gf(R), (2.3) 16πG where the function f(R) of the Ricci curvature R can in general be of any form. In the special case when f(R) = R one recovers the Einstein-Hilbert action which yields the Einstein equations of general relativity, Eq. (1.2). For every dark energy model it is possible to find a function f(R) that will result in exactly the same expansion history [see, e.g. 74–76] thus potentially eliminating the need for dark energy. However, nothing prevents the coexistence of modified gravity and dark energy, with both contributing to powering the current accelerated cosmological expansion. It is of significant importance to be able to determine which scenario best describes what is taking place in our Universe. options see, e.g., [51], [52], [53], [54], [55], [56], [57], and [58]. 3 For constraints on cosmological parameters from data from space missions proposed earlier, see [73] and references therein.

27 This chapter is based on [77]. We investigate how well anticipated data from the galaxy surveys mentioned above can constrain the time dependence of the dark energy. We will use the Fisher matrix formalism to obtain predictions for the φCDM model and compare these with those made using the (model-dependent) XCDM and ωCDM parameterizations of dark energy. We will mostly assume that gravity is well described by general relativity, but will also look at some simple modified gravity cases. We find that the anticipated constraints on the parameter α of the φCDM model are almost an order of magnitude better than the ones that are currently available. Compared to the recent analysis of [70], here we use an updated characterization of planned next-generation space-based galaxy surveys, so our forecasts are a little more realistic. We also consider an additional dark energy parametrization, XCDM, a special case of ωCDM that was considered by [70], as well as the φCDM model, forecasting for which has not previously been done. The rest of the chapter is organized as follows. In Sec. 2.2 we briefly describe the observables and their relationship to basic cosmological parameters. In Sec. 2.3 we describe the models of dark energy that we study. Section 2.4 outlines the method we use for predicting parameter constraints, with some details given in the AppendixA. We present our results in Sec. 2.5 and provide conclusions in Chapter5.

2.2 Measured power spectrum of galaxies

The large-scale structure of the Universe, which most likely originated as quantum-mechanical fluctuations of the scalar field that drove an early epoch of inflation [see, e.g., 78], became (electromagnetically) observable at z ∼ 103 after the recombination epoch. Dark energy did not play a significant role at this early recombination epoch because of its low mass- density relative to the densities of ordinary and dark matter as well as that of radiation (neutrinos and photons). At z ∼ 5 galaxy clusters began to form. Initially, in regions where

28 the matter density was a bit higher than the average, space expanded a bit slower than average. Eventually the dark and ordinary matter reached a minimum density and the regions contracted. If an over-dense region was sufficiently large its baryonic matter collapsed into its dark-matter halo. The baryonic matter continued to contract even more due to its ability to lose thermal energy through the emission of electromagnetic radiation. This can not happen with dark matter since it does not emit significant electromagnetic radiation nor does it interact significantly (non-gravitationally) with baryonic matter. As a result the dark matter remained in the form of a spherical halo around the rest of the baryonic part of a galaxy. At z ∼ 2 the rich clusters of galaxies were formed by gravity, which gathered near-by galaxies together. Also by this time the dark energy’s energy density had become relatively large enough to affect the growth of large-scale structure. Different cosmological models with different sets of parameters can result in the same expansion history and so it impossible to distinguish between such models by using only expansion history measurements. This is one place where measurements of the growth history of the large-scale structure of the Universe plays an important role. It is not possible to fix free parameters of two different cosmological models to give exactly the same expansion and growth histories simultaneously. It is therefore vital to observe both histories in order to obtain better constraints on parameters of a cosmological model. In a cosmological model described by the FLRW metric, and to lowest order in dark matter over-density perturbations, the power-spectrum of observed galaxies is given by [79]

2 2 Pg(k, µ) = Pm(k)(bσ8 + fσ8µ ) . (2.4)

Here subscript g denotes galaxies, Pm is the underlying matter power spectrum, b is the bias of galaxies, f is the growth rate, µ is the cosine of the angle between wave-vector k and the line-of-sight direction, and σ8 is the overall normalization of the power spectrum

−1 (σ8 is the rms energy density perturbation smoothed over spheres of radius 8h Mpc,

−1 −1 where h = H0/(100kms Mpc ) and H0 is the Hubble constant). Since, for a measured

29 power spectrum of galaxies on a single redshift slice, the bias and growth rate are perfectly degenerate with the overall amplitude, in the equations below we will refer to bσ8 and fσ8 simply as b and f. The angular dependence of the power spectrum in Eq. (2.4) can be used to infer the growth rate factor f(z) which is deﬁned as the logarithmic derivative of the linear growth factor d ln G f(z) = , (2.5) d ln a where a is the cosmological scale factor, and the linear growth factor G(t) = δ(t)/δ(tin)

4 shows by how much the perturbations have grown since some initial time tin. The numerical value of the f(z) function depends both on the theory of gravity and on the expansion rate of the Universe. Since the growth rate depends very sensitively on the total amount of non-relativistic matter, it is often parametrized as [see, e.g., 80, and references therein]

γ f(z) ≈ Ωm(z), (2.6) where Ω (1 + z)3 Ω (z) = m , (2.7) m E2(z) and

p 3 2 E(z) = H(z)/H0 = Ωm(1 + z) + Ωk(1 + z) + ΩDE(z). (2.8)

Here H(z) is the Hubble parameter and H0 is its value at the present epoch (the Hubble constant), Ωm is the value of the energy density parameter of non-relativistic matter at the present epoch (z = 0), Ωk that of spatial curvature, and ΩDE(z) is the energy density parameter which describes the evolution of the dark energy density and is diﬀerent in diﬀerent dark energy models.

4 Here we have expanded the energy density ρ(t, x) in terms of a small spatially inhomogeneous fractional perturbation δ(t, x) about a spatially-homogeneous background ρb(t): ρ(t, x) = ρb(t)[1 + δ(t, x)].

30 The growth index, γ, depends on both a model of dark energy as well as a theory of gravity. When general relativity is assumed and the equation of state of dark energy is taken to be of the general form in Eq. (2.2) then [see, e.g., 80, and references therein]

γ ≈ 0.55 + 0.05[1 + ω(z = 1)] (2.9) to a few percent accuracy. In the ΛCDM cosmological model γ ≈ 0.55. An observed significant deviation from this value of γ will present a serious challenge for the standard cosmological model. The power spectrum is measured under the assumption of a fiducial cosmological model. If the angular and radial distances in the fiducial model differ from those in the real cosmology, the power spectrum will acquire an additional angular dependence via the Alcock- Paczyński(AP) [81] effect, as discussed in [70],

2 2 1 k p 2 2 2 µ f Pg(k, µ) = 2 Pm F + µ (1 − F ) × b + 2 2 2 , (2.10) fkf⊥ f⊥F F + µ (1 − F ) where ˆ fk(z) = Rr(z)/Rr(z), (2.11)

ˆ f⊥(z) = DA(z)/DA(z), (2.12)

F = fk/f⊥. (2.13)

Here Rr = dr/dz is the derivative of the radial distance, DA is the angular diameter distance (both defined below), a hat indicates a quantity evaluated in the fiducial cosmological model, and a quantity without a hat is evaluated using the alternative cosmological model. The AP effect is an additional source of anisotropy in the measured power spectrum and allows for the derivation of stronger constraints on cosmological parameters.

31 2.3 Cosmological models

In an FLRW model with only non-relativistic matter and dark energy the distances DA(z) and Rr(z) are 1 Z z dz0 √ p DA(z) = sinh Ωk 0 , (2.14) h Ωk(1 + z) 0 E(z ) 1 R (z) = . (2.15) r h(1 + z)E(z)

Here E(z) is deﬁned in Eq. (2.8). The functional form of E(z) depends on the model of dark energy.

2.3.1 ΛCDM, XCDM and ωCDM parameterizations

Here we describe the relevant features of the ΛCDM model and the dark energy parameterizations we consider. If the dark energy is taken to be a ﬂuid its equation of state can be written as p = ω(z)ρ. For the ΛCDM model the equation of state parameter ω(z) = −1 and the dark energy density is time independent.

In the XCDM parametrization ω(z) = ωX (< −1/3) is allowed to take any time-independent value, resulting in a time-dependent dark energy density. In the ωCDM parametrization the time dependence of ω(z) is parametrized by intro- ducing an additional parameter ωa through [82, 83]

z w(z) = w + w . (2.16) 0 a 1 + z

The XCDM parametrization is the limit of the ωCDM parametrization with ωa = 0. In the

ωCDM parametrization the function ΩDE(z) that describes the time evolution of the dark

32 energy density is

z Ω (z) = (1 − Ω − Ω )(1 + z)3(1+w0+wa) exp −3w , (2.17) DE m k a 1 + z

and the corresponding expression for the XCDM case can be derived by setting ωa = 0 here.

2.3.2 φCDM model

In the φCDM model the energy density of the background, spatially homogeneous, scalar ﬁeld φ can be found by solving the set of simultaneous ordinary diﬀerential equations of motion, a˙ φ¨ + 3 φ˙ + V 0(φ) = 0, (2.18) a a˙ 2 8πG k = (ρ + ρ ) − , (2.19) a 3 φ a2

1 1 ρ = φ˙2 + V (φ) . (2.20) φ 16πG 2

Here an over-dot denotes a derivative with respect to time, a prime denotes one with respect to φ, V (φ) is the potential energy density of the scalar field, ρφ is the energy density of the scalar field, and ρ that of the other constituents of the Universe. Following [7] we consider a scalar field with inverse-power-law potential energy density

κ V (φ) = φ−α. (2.21) 2G

Here α is a positive parameter of the model to be determined experimentally and κ is a positive constant. This choice of potential has the interesting property that the scalar ﬁeld solution is an attractor with an energy density that slowly comes to dominate over the energy density of the non-relativistic matter (in the matter dominated epoch) and causes

33 the cosmological expansion to accelerate. The function ΩDE(z) in the case of φCDM is

1 κ Ω (z) = φ˙2 + φ−α . (2.22) DE 12 G

2.4 Fisher matrix formalism

The precision of the galaxy power spectrum measured in redshift bins depends on the cosmological model, the volume of the survey, and the distribution of galaxies within the observed volume. See AppendixA for a summary of how to estimate the precision of measurements from survey parameters.

meas We assume that the power spectrum P (ki) has been measured in N wave-number ki bins (i = 1 ...N) and each measurement has a Gaussian uncertainty σi. From these measurements a likelihood function

1 L ∝ exp − χ2 (2.23) 2 can be constructed where N meas 2 X (P − Pi(p)) χ2 = i . (2.24) σ2 i=1 i Here p are the set of cosmological parameters on which the power spectrum depends. The likelihood function in Eq. (2.23) can be transformed into the likelihood of theoretical parameters p by Taylor expanding it around the maximum and keeping terms of only second

2 j k 5 order in δp as χ (δp) = Fjkδp δp , where Fjk is the Fisher matrix of the parameter set p given by second derivatives of the likelihood function through

∂2 ln L F = − . (2.25) jk ∂pj∂pk

5 For a review of the Fisher matrix formalism as applied to cosmological forecasting, see [84].

34 The Fisher matrix predictions are exact in the limit where initial measurements as well as derived parameters are realizations of a Gaussian random variable. This would be the case

meas if the Pi were perfectly Gaussian and the Pi(p) were linear functions of p, which would make the second order Taylor expansion of the likelihood around its best fit value exact. In reality, because of initial non-Gaussian contributions and nonlinear effects, the predictions of Fisher matrix analysis will be different (more optimistic) from what is achievable in practice. These differences are larger for strongly non-linear models and for the phase spaces in which the likelihood is non-negligible at some physical boundary (α = 0 in case of φCDM). A more realistic approach, that requires significantly more computational time and power, is to generate a large amount of mock data and perform a full Monte-Carlo Markov Chain (MCMC) analysis [see, e.g., 85, 86, where the authors find significant differences compared to the results of the Fisher matrix analysis]. We assume that the full-sky space-based survey will observe Hα-emitter galaxies over 15000 deg2 of the sky. For the density and bias of observed galaxies we use predictions from [87] and [88] respectively. We further assume that about half of the galaxies will be detected with a reliable redshift. These numbers roughly mirror what proposed space missions, such as the ESA Euclid satellite and the NASA WFIRST mission, are anticipated to achieve. For the fiducial cosmology we use a spatially-flat ΛCDM model with Ωm = 0.25, the baryonic matter density parameter Ωb = 0.05, σ8 = 0.8, and the primordial density perturbation power spectral index ns = 1.0, for convenience we summarize all the parameters of the fiducial model in Table 1. We further assume that the shape of the power spectrum is known perfectly (for example from the results of the Planck satellite) and ignore derivatives of the real-space power spectrum with respect to cosmological parameters. We predict the precision of the measured galaxy power spectrum and then transform it into correlated error bars on the derived cosmological parameters. At first we make predictions for the basic quantities b and f in the XCDM and ωCDM parameterizations

35 and in the φCDM model. Then it allows us to predict constraints on deviations from general relativity and see how these results change with changing assumptions about dark energy. Finally, we forecast constraints on the basic cosmological parameters of dark energy models. For the XCDM parametrization these basic cosmological parameters are pXCDM = (f, b, h, Ωm, Ωk, wX ). The ωCDM parametrization has one extra parameter de- scribing the time evolution of the dark energy equation of state parameter, pωCDM =

(f, b, h, Ωm, Ωk, w0, wa). For the φCDM model the time dependence of the dark energy density depends only on one parameter α so we have pφCDM = (f, b, h, Ωm, Ωk, α). In order to derive constraints on the parameters of the considered cosmological models while altering assumptions about the correctness of general relativity, we transform Fisher ma- trices of each model from the parameter set described above to the following parameter set (that now includes γ that parametrizes the growth rate) pmodel = (γ, model), where by model we mean all the parameters of a particular model, for example, for ωCDM model = pωCDM = (f, b, h, Ωm, Ωk, w0, wa).

2.5 Results

2.5.1 Constraints on growth rate

Figure 3.1 shows predictions for the measurement of growth rate assuming different dark energy models. We find that in the most general case, when no assumption is made about the nature of dark energy, the growth rate can be constrained to a precision of better then 2% over a wide range of redshifts. This is in good agreement with previous similar studies [see, e.g., Fig. 1 of 70]. When we specify a dark energy model the constraints on growth rate improve by about a factor of two. There is very little difference between the results derived for different dark energy models: the precision is almost insensitive to the assumed model. Also, one can notice that the curves for the XCDM parametrization and for the φCDM

36 model are almost identical. The likely explanation of this effect is that for a fixed redshift bin the φCDM model is well described by the XCDM parametrization with the value of the parameter ωX = pφ/ρφ, where the values of the scalar field pressure pφ and energy density

ρφ are evaluated at that redshift bin. The measurements of growth rate can be remapped into constraints on parameters de- scribing the deviation from general relativity. Figure 3.2 shows correlated constraints between the current re-normalized Hubble constant h and the γ parameter that describes the growth of structure. The φCDM model constraints on both h and γ are tighter than those for the XCDM or ωCDM parameterizations. As expected, the most restrictive ΛCDM model results in the tightest constraints. In Table 2.2 we show the deviations of the parameter γ from its ﬁducial value for various dark energy models.

2.5.2 Constraints on dark energy model parameters

We use measurements of growth and distance to constrain parameters of the dark energy models. Figure 3.3 shows constraints on parameters of the ωCDM parameterization [these should be compared to Figs. 4a and 5a of 70] . When no assumptions are made about the nature of gravity the constraints on ω0 and ωa are very weak and degenerate. When we assume general relativity the constraints tighten signiﬁcantly, resulting in ∼ 10% accuracy in the measurement of ω0 and ∼ 25% accuracy in the measurement of ωa.

The upper panel of Fig. 3.4 shows constraints on the parameters ωX and Ωm of the XCDM parametrization. Similar to the previous case, the constraints tighten signiﬁcantly when we assume general relativity as the model of gravity. About a 2% measurement of ωX and a 5% measurement of Ωm are possible in this case. The lower panel of Fig. 3.4 show the related constraints on Ωk and Ωm for the XCDM parametrization. The constraints are similar to, but somewhat tighter than, those for the ωCDM parametrization. This is because the XCDM parametrization has one less parameter than the ωCDM parametrization. Spatial

37 curvature can be constrained to about 15% precision in this case. Figures 2.5 and 2.6 show constraints on parameters of the φCDM model. In the most general case, when no assumption is made about the nature of gravity, the constraints are weak and the parameters α and Ωm are strongly correlated, with larger values of α requiring larger values of Ωm. When general relativity is assumed, the constraints become much stronger and parameter α can be constrained to be less than 0.1 at the 1-σ confidence level. This is significantly better than any constraint available at the moment. Figure 2.7 shows constraints on the parameters of the ΛCDM model. From the clustering data alone the spatial curvature can be constrained with almost 1% precision, largely because this model has the least number of free parameters. The exact numerical values for the forecast error bars and likelihood contours should be taken with caution and not be interpreted as predictions for the performance of any specific survey (such as Euclid or WFIRST). Our main objective in this work was first to investigate how the modified gravity constraints change with different models of dark energy and second to demonstrate the improvement in φCDM model constraints achievable with future galaxy surveys. Because of this we were able to simplify our method by adopting a Fisher matrix formalism instead of a full MCMC approach and also use a simplified description of the survey baseline. For more realistic predictions of Euclid performance, see, e.g., [67, 70, 71].

38 Table 2.1: Values of the parameters of the ﬁducial ΛCDM model and the survey. 2 Ωm Ωb Ωk h σ8 ns Eﬃciency Redshift span Covered sky area in deg 0.25 0.05 0.0 0.7 0.8 1.0 0.45 0.55 ≤ z ≤ 2.05 15000

Table 2.2: Predicted deviations of parameter γ from its fiducial value, at one standard deviation confidence level, for different assumptions about dark energy. DE model Fiducial γ deviation ωCDM 0.55 0.035 φCDM 0.55 0.023 XCDM 0.55 0.035 ΛCDM 0.55 0.016

0.025

Free Geometry

wCDM

CDM

XCDM

0.020

CDM )/f

0.015

0.010

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Figure 2.1: Predicted relative error on the measurements of growth rate as a function of redshift z in redshift bins of ∆z = 0.1 for diﬀerent models of dark energy. The upper solid black line shows predictions for the case when no assumption is made about the nature of dark energy.

0.706

wCDM

CDM

XCDM

0.704

CDM

0.702

0.700

0.698

0.696

0.694

0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59

Figure 2.2: Predicted one standard deviation conﬁdence level contour constraints on the current renormalized Hubble constant h and the parameter γ that describes deviations from general relativity for diﬀerent dark energy models.

1.25

Free Growth

1.00

0.75

0.50

0.25

0.00 a

-0.25

-0.50

-0.75

-1.00

-1.25

-1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7

Free Growth

0.02

0.01

0.00 k

-0.01

-0.02

0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

Figure 2.3: Upper panel shows one standard deviation conﬁdence level contours constraints on parameters ωa and ω0 of the ωCDM parametrization, while lower panel shows these for parameters Ωk and Ωm.

-0.92

Free Growth

-0.94

-0.96

-0.98

-1.00 x

-1.02

-1.04

-1.06

-1.08

0.240 0.245 0.250 0.255 0.260

0.020

Free Growth

0.015

0.010

0.005

0.000 k

-0.005

-0.010

-0.015

-0.020

0.240 0.245 0.250 0.255 0.260

Figure 2.4: One standard deviation conﬁdence level contour constraints on parameters of the XCDM parametrization.

1.75

Free Growth

GR 1.50

1.25

1.00

0.75

0.50

0.25

0.00

0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34

0.11

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

0.242 0.244 0.246 0.248 0.250 0.252 0.254 0.256 0.258

Figure 2.5: One standard deviation conﬁdence level contour constraints on parameters α and Ωm of the φCDM model. Lower panel shows a magniﬁcation of the tightest two contours in the lower left corner of the upper panel.

0.035

0.030

Free Growth

0.025

0.020

0.015

0.010

0.005

0.000 k

-0.005

-0.010

-0.015

-0.020

-0.025

-0.030

-0.035

0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350

0.020

0.015

0.010

0.005

0.000 k

-0.005

-0.010

-0.015

-0.020

0.244 0.246 0.248 0.250 0.252 0.254 0.256

Figure 2.6: One standard deviation conﬁdence level contour constraints on parameters Ωk and Ωm of the φCDM model. The lower panel shows a magniﬁcation of the two tightest contours in the center of the upper panel.

0.020

Free Growth

0.015

0.010

0.005

0.000 k

-0.005

-0.010

-0.015

-0.020

0.242 0.244 0.246 0.248 0.250 0.252 0.254 0.256 0.258

Figure 2.7: One standard deviation conﬁdence level contour constraints on parameters Ωk and Ωm of the ΛCDM model.

46 Chapter 3

Nonﬂat time-variable dark energy cosmology

3.1 Introduction

When measurements of the cosmic microwave background (CMB) anisotropy are examined in the context of the current standard model of cosmology, the ΛCDM model1, they indicate that the cosmological spatial hypersurfaces are close to flat, see [1, 92, 93]. On the other hand, under the assumption of flat spatial geometry the data favor time-independent dark energy (DE). However, it has been known for a while now that if a spatially curved time-variable DE model is used to analyze the CMB anisotropy measurements there is a degeneracy between spatial curvature and the parameter that governs the DE time variability, and this results in significantly weaker constraints on both parameters compared to the cases when only either non-zero spatial curvature or DE time variability is assumed, early

1In this model [6], the current cosmological energy budget is dominated by a cosmological constant Λ, with non-relativistic cold dark matter (CDM) being the next largest contributor. For some time now most observations have been reasonably consistent with the predictions of the spatially-ﬂat ΛCDM model; For early indications, see e.g., [35, 47, 89, 90]. Note that there are tentative observational indications that the standard CDM structure formation model, assumed in the ΛCDM cosmological model, might need to be improved upon [91].

47 work includes [94–107]. Most of these analyses are based on the XCDM parametrization or generalizations thereof. In the XCDM parametrization time-evolving DE is taken to be an X-fluid with equation of state pX = wX ρX where ρX and pX are the X fluid energy density and pressure and the equation of state parameter wX < −1/3 is a constant. This is an incomplete model of time-variable DE since, unless extended, it cannot consistently describe the evolution of spatial inhomogeneities [43] and [44]. The ΦCDM model [7, 45] is the simplest consistent model of time-variable DE. In this model a scalar field Φ with potential energy density V (Φ) is the DE; V (Φ) ∝ Φ−α, where constant α > 0, is a widely used example2. The original ΦCDM model assumed flat spatial sections. In this paper we develop the curved space extension of the ΦCDM model. Related models have been previously considered, see Ref. [94–96, 109–111]. However, as far as we are aware, we are the first to establish that the scalar field solution in the curvature-dominated epoch is a time-dependent fixed point or attractor, and that in the curvature-dominated epoch the scalar field energy density grows relative to that of space curvature, generalizing the results of [7, 45] to curved space. This chapter is based on [112]. In the next section we describe the curved-space ΦCDM model we study. In this section and in the AppendixB we show that this model has a time-dependent fixed point scalar field solution in the curvature-dominated epoch. In Sec. 3.3 we compute some observable cosmological-test predictions for this model as a function of the three cosmological parameters of the model. Then we discuss these results by comparing those for flat and nonflat geometries as well as for open and closed geometries. In Chapter 5 we provide conclusions.

2See Ref. [108] for more general examples.

48 3.2 The Model

The original ΦCDM model of [7] was designed to describe the late-time consequences of an inflationary scalar field Φ model in which the scalar field potential energy density V (Φ) has an inverse power-law tail at large Φ. This form of V (Φ) was chosen because it provides a self-consistent phenomenological description of DE whose density decreases as the Universe expands, but decreases less rapidly than the nonrelativistic (cold dark and baryonic) matter density in a spatially flat universe. This eventually results in the expansion reaching a point at which the densities of nonrelativistic matter and DE have the same value and the decelerating cosmological expansion of the matter-dominated epoch switches to the accelerating expansion of the DE-dominated epoch that is currently observed [113–115].

In spacetime coordinates xµ (µ = 0, 1, 2, 3), with units chosen so that ~ = c = 1, the late-time action of the model we consider is

Z √ m 2 1 S = d4x −g p −R + gµν∂ Φ∂ Φ 16π 2 µ ν κ − m 2Φ−α + L . (3.1) 2 p

−1/2 Here the Planck mass mp = G where G is the gravitational constant and L is the Lagrangian density of ordinary matter. The constants κ and α are positive real numbers and we adopt

8 α + 4 2 α/2 κ = α(α + 2) . (3.2) 3 α + 2 3

With this choice for κ, our results in the limit of zero space curvature reduce to those of Ref. [7]. Applying the variational principle with respect to the metric to the action (3.1) gives

49 the Einstein equations,

1 8π Rµν − Rgµν = 2 (Tµν + Qµν) . (3.3) 2 mp

Here Rµν and R are the Ricci tensor and scalar and Tµν is the stress-energy tensor of ordinary matter while Qµν is the stress-energy tensor of the Φ ﬁeld and has the form

m 2 Q = p 2∂ Φ∂ Φ − gζξ∂ Φ∂ Φ − κΦ−α g . µν 32π µ ν ζ ξ µν (3.4)

Assuming the cosmological principle of large-scale spatial homogeneity, the Friedmann metrics in coordinates (t, r, θ, ϕ) are

dr2 ds2 = dt2 − a2 + r2dθ2 + r2 sin2 θdϕ2 . (3.5) 1 − kr2

Here a is the scale factor and k is the curvature parameter that takes values −1, 0, 1 for open, flat, and closed spatial geometry. References [7, 45] consider only the k = 0 case. The equation of motion for the scale factor a can be obtained by substituting the metric of Eq. (3.5) into the Einstein equations (3.3). The equation of motion for the scalar field Φ can be derived by either applying the variational principle with respect to the Φ field to the scalar field part of the action (3.1), or from the continuity conditions on the scalar field stress-energy tensor Qµν given in Eq. (3.4), and then using the metrics of Eq. (3.5). The complete system of equations of motion is

a˙ κα Φ¨ + 3 Φ˙ − m 2Φ−(α+1) = 0, (3.6) a 2 p a˙ 2 8π k = 2 (ρ + ρΦ) − 2 , a 3mp a m 2 ρ = p Φ˙ 2 + κm 2Φ−α . Φ 32π p

50 Here an overdot denotes a derivative with respect to time, and ρ is the energy density of ordinary matter while ρΦ is that of the dark energy scalar ﬁeld Φ. It is also useful to introduce the density of spatial curvature,

3m 2 k ρ = − p . (3.7) k 8π a2

In this convention the spatially open model has ρk > 0.

DE cannot have a significant effect at early times, so we assume ρΦ ρ at a(t) a0, where a0 is the current value of the scale factor. Neither can space curvature play a significant role in the early nonrelativistic matter-dominated epoch, so ρk ρ for a(t) a0. Under these assumptions the Einstein–de Sitter model provides an accurate description of the nonrelativistic matter-dominated epoch and so can be used to derive initial conditions for the scalar field Φ identical to these in the original flat-space case of Ref. [7]. Of course, since the solution is a time-dependent fixed point or attractor, as shown here and in Refs. [7, 45], it is not sensitive to the precise initial conditions adopted: a large range of initial conditions results in the same scalar field fixed point or attractor solution.

3.2.1 Solution for the curvature-dominated epoch

In order to ﬁnd whether the system (3.6) has an attractor solution in a certain epoch (i.e. matter dominated, radiation dominated or curvature dominated) we use a perturbation theory approach in which we treat the energy density of the scalar ﬁeld Φ as a perturbation. Therefore, we neglect all terms in the right-hand side of the second equation of the system (3.6) (i.e. the Friedmann equation) except the energy density which dominates at the epoch of interest. When the energy budget of the Universe is dominated by radiation, ordinary matter or curvature, the solution of the Friedmann equation for the scale factor a varies as a power of time, a ∝ tn (which in general is not true in a quintessence-dominated epoch), where the index n is determined for each epoch (as discussed in more detail later in this

51 section). By substituting this power-law solution for the scale factor a ∝ tn into the system (3.6), the equation of motion for the scalar ﬁeld is

3n κα Φ¨ + Φ˙ − m 2Φ−(α+1) = 0. (3.8) t 2 p

Equation (3.8) has a special power-law solution

2/(α+2) Φe(t) = At (3.9) where the label e denotes that this is an unperturbed, exact, spatially homogeneous solution. The value of the constant A is

καm 2(α + 2)2 1/(α+2) A = p . (3.10) 4[3n(α + 2) − α]

We now show that, for the range of α and n values that we are interested in, the special solution (3.9) is an inwardly spiraling attractor in the phase space of solutions to (3.8). This means, for example, that in a curvature-dominated epoch (which has n = 1), the scalar ﬁeld will approach the special solution (3.9) for a wide range of initial conditions. In order to show this we follow the methods of Sec. V of Ref. [45], and make the change of variables (Φ, t) 7→ (u, τ) where

τ Φ(t) = Φe(t)u(t), t = e . (3.11)

Substituting (3.11) into (3.8) and using (3.9) for Φe(t) we derive the equation for perturbation u(t) of the scalar ﬁeld Φ(t),

4 6n(α + 2) − 2α u00 − 1 − 3n − u0 + u − u−(α+1) = 0. (3.12) α + 2 (α + 2)2

Here primes denote derivatives with respect to τ. Finally we switch to the phase space of

52 solutions of the system (3.8) by rewriting (3.12) as the system

u0 = p, 4 6n(α + 2) − 2α p0 = 1 − 3n − p − u − u−(α+1) . (3.13) α + 2 (α + 2)2

The critical point (u0, p0) = (1, 0) corresponds to the special solution (3.9). Although there exist, in general, other critical points at p = 0, these involve complex roots of unity for u, which are not physically relevant in this case. Taking the linearization of (3.13) about the critical point, one obtains the eigenvalues

p λ1,2 = f(α, n) ± i g(α, n) (3.14) where

α − 2 − 3n(α + 2) f(α, n) = , (3.15) 2(α + 2) 6n(α + 2)(5α + 6) − 9n2(α + 2)2 − (3α + 2)2 g(α, n) = . 4(2 + α)2

For f(α, n) < 0 and g(α, n) > 0, the eigenvalues λ1 and λ2 show that the critical point is an inwardly spiraling attractor in the phase space. The cases n = 1/2 (radiation-dominated epoch) and n = 2/3 (matter-dominated epoch) were previously studied in Ref. [45]. Note that for the case n = 1 (curvature-dominated epoch) our critical point is an inwardly √ spiraling attractor if α > −2 + 2/ 3 or if α < −4. In the ΦCDM model we are specifically interested in the range α > 0. So the critical point is an attractor for all α values of interest. The above analysis ignores spatial inhomogeneities in the gravitational field. In the Appendix we show that the time-dependent fixed point solution found above remains stable in the presence of gravitational field inhomogeneities. We can use our results to show how this model partially resolves the “coincidence” puzzle.

53 From the last equation of the system (3.6) it follows that in the curvature-dominated epoch

−2α/(α+2) ρΦ(t) ∝ t , (3.16)

2 3 while ρk(t) ∝ 1/t and ρm(t) ∝ 1/t . The exponent in Eq. (3.16) varies from −2 to 0 as α varies from ∞ to 0, thus for α < ∞ ρΦ(t) decays at a slower rate than ρk in the curvature- dominated epoch and eventually comes to dominate. This is consistent with the results of similar analyses in the radiation-dominated and matter-dominated epochs given in Ref. [45].

3.3 Some observational predictions

To gain some insight into the eﬀects space curvature has on the ΦCDM model, we compute predictions for some cosmological tests in this section. To make these predictions we ﬁrst numerically integrate the equations of motion (3.6) with initial condition of the form (3.9) taken in the matter-dominated epoch, where n = 2/3 with the usual expression for the scale factor in the matter-dominated epoch, see Ref. [7]. Instead of ρ, ρΦ and ρk we use dimensionless density parameters such as

8πρ ρ Ωm = = , (3.17) 2 2 3mp H ρ + ρk + ρΦ where H =a/a ˙ is the Hubble parameter. We present the predictions as isocontours in the space of model parameters (Ωm0,α) for a number of diﬀerent values of the spatial curvature density parameter Ωk0. (Here the subscript 0 refers to the value at the current epoch. For the open model Ωk0 > 0.) For our illustrative purposes here we consider the same four cosmological tests studied in Ref. [7]. For a discussion of these and other cosmological tests see Ref. [41]. While it is of great interest to determine constrains on the three cosmological parameters of the model — Ωm0,Ωk0, and α — using various cosmological observables, in this paper we restrict ourselves to some qualitative remarks; a detailed quantitative comparison

54 between the predictions of the model and observations is given in Ref. [116], where it is found that observational data less tightly constrains space curvature in a dynamical dark energy matter of the type we study here.

3.3.1 The time parameter H0t0

The dimensionless time parameter is

Z a0 da H0t0 = H0 , (3.18) 0 a˙(t)

where t0 is the age of the Universe and H0 and a0 are the present values of the Hubble parameter and scale factor. Figure 3.1 shows contours of constant H0t0 as a function of Ωm0 and α

−1 −1 for a series of ﬁxed values of Ωk0. A recent summary estimate of H0 = 68±2.8 km s Mpc

0.041 [117] and the Planck (with WMAP polarization) estimate of t0 = 13.824± 0.055 Gyr [1, 92, 93] gives, for the 2σ range, 0.88 ≤ H0t0 ≤ 1.04, where we have added the 1σ errors in quadrature and doubled to get the 2σ range. From Fig. 3.1 we see that Ωm0 = 0.27 and α = 3 is reasonably consistent with these constraints for a range of Ωk0. In the limit α → 0 this model reduces to the constant Λ one (but not necessarily with zero space curvature), while the limit α → ∞ corresponds to the open, closed, or ﬂat (Einstein–de Sitter) model with Λ = 0, depending on the value of space curvature. At ﬁxed

Ωm0 (and Ωk0), or in the flat-space case [7], the effect of increasing α is to reduce the value of H0t0, making the Universe younger at fixed H0, since α = 0 corresponds to a constant Λ and so the oldest Universe for given Ωm0 and Ωk0. However, nonzero space curvature brings interesting new effects. At α = 0 the ΦCDM model reduces to the ΛCDM one and here it is well known that to hold H0t0 constant in the open case as Ωm0 is reduced and Ωk0 is increased requires a decrease in ΩΛ (to compensate for the increase of t0 at constant H0 as

Ωm0 is reduced and Ωk0 is increased). The converse is true in the closed case. Studying the

α = 0 intercepts of the H0t0 isocontours in both panels of Fig. 3.1 conﬁrms these arguments.

55 8

Open 0.20

Open 0.10

Flat

Closed 0.10

Closed 0.20

0.85

0.95

5 0.8

0.7

0.75

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.80

0.85 0.90

0.95

Open 0.05

Open 0.02

Flat

Closed 0.02

Closed 0.05

0.0 0.1 0.2 0.3 0.4 0.5

Figure 3.1: Contours of ﬁxed time parameter H0t0, as a function of the present value of the nonrelativistic matter density parameter Ωm0 and scalar ﬁeld potential power-law index α, at various values of the current value of the space-curvature density parameter Ωk0 (as listed in the inset legend boxes). The upper panel shows a larger part of (Ωm0, α) space for a larger range of Ωk0 values [for H0t0 = 0.7, 0.75, 0.8, 0.85, 0.95, 1.05 and 1.15, from right to left], while the lower panel focuses on a smaller range of the three parameters [for H0t0 from 0.8 to 1.15 in steps of 0.05, from right to left].

56 That is, for a fixed value of H0t0 at smaller α (i.e. α . 4) the contours corresponding to open geometry shift to the left of the flat geometry, i.e., to lower Ωm0, while the contours corresponding to closed geometry shift to the right of the flat case. At higher α the DE density deceases more rapidly with the expansion (unlike the α =

0 case where Λ remains constant), and the contours switch around. Here to hold H0t0 constant in the open case as Ωk0 is increased requires that Ωm0 increase and ΩΦ0 decrease to compensate. In the closed case as Ωk0 is increased, Ωm0 must decrease and ΩΦ0 must also decrease. Thus, as evident from Fig. 3.1, for a given H0t0 value there is a point in (Ωm0, α) space at which contours corresponding to diﬀerent space curvatures cross. The intersection point moves to larger α as Ωm0 is decreased. This is because the Universe is older (at ﬁxed

H0) at smaller Ωm0 so even DE with larger α now has more time to come to dominate the energy budget (and so behave more like DE with a constant DE density).

3.3.2 The distance modulus diﬀerence ∆m(z)

We next consider the diﬀerence in bolometric distance moduli, at redshift z = 1.5, of the ΦCDM model and the Einstein–de Sitter model. The coordinate distance r is

1 p Z t0 dt r = √ sin −Ωk0 . (3.19) −Ωk0 tem a(t)

Here tem and t0 are the times when the signal was emitted and received. Thus the diﬀerence in the distance moduli of the two models is

r ∆m(z) = 5 log10 (3.20) rEdS

where rEdS is the coordinate distance in the Einstein–de Sitter model.

Figure 3.2 shows contours of constant ∆m(z = 1.5) as a function of Ωm0 and α for some values of Ωk0. Comparing Figs. 3.1 and 3.2, we see that near α = 0, where the DE behaves

57 8

Open 0.20

Open 0.10

Flat

Closed 0.10

Closed 0.20

0.8 0.6

0.5

0.4 0.3 0.1

0.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.3

0.4

0.5

1 0.6

0.7

Open 0.05

0.8

Open 0.02

Flat

Closed 0.02

Closed 0.05

0.0 0.1 0.2 0.3 0.4 0.5

Figure 3.2: Contours of fixed bolometric distance modulus relative to the Einstein–de Sitter model, ∆m(z = 1.5), as a function of the matter density parameter Ωm0 and scalar field potential power-law index α, and various values of the space curvature density parameter Ωk0 (as listed in the inset legend boxes). The upper panel shows a larger part of (Ωm0, α) space for a larger range of Ωk0 values [for ∆m(z = 1.5) = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.8 from right to left], while the lower panel focuses on a smaller range of the three parameters [for ∆m(z = 1.5) from 0.3 to 0.8 in steps of 0.1, from right to left]. In the upper panel there is no Ωk0 = 0.2 contour for ∆m(z = 1.5) = 0.1 since in this case the model is too open for such a small distance modulus difference. 58 like constant Λ, ∆m(z = 1.5) is less sensitive to the value of Ωk0 than is H0t0. However at larger α ∆m(z = 1.5) is more sensitive to spatial curvature than is H0t0. Clearly, extending the ΦCDM model to include space curvature as a free parameter broadens the range of allowed parameter values. As in the H0t0 case, for a given value of ∆m(z = 1.5) there is a point in (Ωm0, α) space at which all contours intersect.

3.3.3 Number counts

The count per unit increment of redshift for conserved objects is3

dN H 3a 2r2 a ∝ z2A(z),A(z) = 0 0 . (3.21) dz z2 a˙

Isocontours of ﬁxed A(z = 0.7) are shown in Fig. 3.3. The general features are similar to those shown in Figs. 3.1 and 3.2 for H0t0 and ∆m(z = 1.5).

3.3.4 The growth of structure

Finally, we consider the growth of large-scale structure of the Universe which started as small primordial density inhomogeneities in the early Universe, see [78, 121], and references therein. Within the framework of linear perturbation theory the scalar ﬁeld stays homogeneous as we show in the AppendixB on the scales of matter perturbations and the density contrast in ordinary matter, δ = δρ/ρ, satisﬁes

¨ a˙ ˙ 4π δ + 2 δ − 2 ρδ = 0. (3.22) a mp

Following Ref. [7] the cosmological test parameter we consider is

δ(t0) ∆(Ωm0, Ωk0, α) = (3.23) (1 + zi)δ(ti)

3See Sec. IV.B.5 of Ref. [41] and Refs. [118–120] for discussions of this test.

Open 0.20

Open 0.10

Flat 7

Closed 0.10

Closed 0.20

0.35

0.25

0.3

0.45

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.30

0.35

0.40

0.45

Open 0.05

Open 0.02

Flat

Closed 0.02

Closed 0.05

0.0 0.1 0.2 0.3 0.4 0.5

Figure 3.3: Contours of ﬁxed A(z = 0.7) as a function of Ωm0 and α at various values of Ωk0 (as listed in the inset legend boxes). The upper panel shows a larger part of the parameter space for A(z = 0.7) = 0.25, 0.3, 0.35 and 0.45 from right to left. The lower panel shows a smaller range of the three parameters for A(z = 0.7) = 0.3, 0.35, 0.4 and 0.45 from right to left.

0.3

0.5

5 0.7

0.9

Open 0.20

1 Open 0.10

Flat

Closed 0.10

Closed 0.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Open 0.05

Open 0.02 0.6

0.4 0.3

Flat 0.2 0.5

Closed 0.02

Closed 0.05

0.7

0.8

0.0 0.1 0.2 0.3 0.4 0.5

Figure 3.4: Contours of the factor by which the growth of ordinary matter perturbations falls below that of the Einstein–de Sitter model, ∆(Ωm0, Ωk0, α), as a function of the matter density parameter Ωm0 and scalar ﬁeld potential power-law index α, and various values of the space curvature density parameter Ωk0 (as listed in the inset legend boxes). The upper panel shows a larger part of (Ωm0, α) space for a larger range of Ωk0 values [for ∆(Ωm0, Ωk0, α) = 0.3, 0.5, 0.7 and 0.9 from left to right], while the right panel focuses on a smaller range of the three parameters [for ∆(Ωm0, Ωk0, α) from 0.2 to 0.8 in steps of 0.1, from left to right]. In the lower panel there is no Ωk0 = 0.2 contour for ∆ = 0.9 since in this case the model is too open to allow such a large growth factor.

61 where t0 denotes the current epoch while ti is the time when the scale factor ai a0, well within the matter-dominated epoch when the Einstein–de Sitter model was a good approximation. Thus the factor ∆ is the ratio by which the growth of linear ﬂuctuations in density have declined below that of the Einstein–de Sitter model prediction. We graph contours of ∆ in Fig. 3.4. There are two interesting facts about the ∆ contours shown in Fig. 3.4. First, the growth rate is quite sensitive to the value of Ωk0, much more so than any of the other parameters we have considered. (This is not unexpected, as it is well known in more conventional models that the growth factor is much more sensitive to Ωm0 when Ωk0 is nonzero.) Second, the curvature dependence of the isocontours is the opposite of that for the other three parameters. So a joint analysis of growth factor and geometry measurements would seem to be a very good way to constrain Ωk0.

62 Chapter 4

Cosmological constraints from large-scale structure growth rate measurements

4.1 Introduction

The following chapter is based on [122]. We assume that GR provides an adequate model for cosmological gravity and we test various models of dark energy (DE) as a possible explanation of the observed accelerated cosmological expansion. In particular, we consider three models of DE. The ﬁrst one is the standard ΛCDM cosmology in which the energy density of DE does not evolve in time and its equation of state (EoS) is pΛ = −ρΛ, where pΛ is the pressure and ρΛ the energy density of DE. Space sections are not assumed to be ﬂat in this case and the cosmological parameters that characterize the model are p = (Ωm0, ΩΛ) where

Ωm0 is the current value of the nonrelativistic CDM and baryonic matter density parameter and ΩΛ is that of Λ. The second model we consider is the simplest modiﬁcation of ΛCDM cosmology in which the energy density of DE is time dependent and its EoS is parametrized as pX = wX ρX , where wX is constant and < −1/3. The upper limit of −1/3 is a consequence

63 of the requirement that DE provide positive acceleration. This spatially-flat XCDM model is the simplest parametrization of dynamical DE, with parameters p = (Ωm0, wX ). However, it is incomplete as it cannot describe density inhomogenities [see, e.g., 44]. The last model we study is the consistent quintessence model of DE in which DE is a scalar field. In particular we consider the much studied spatially-flat φCDM model [7, 45] whose equations of motion in units where ~ = c = 1 are

a˙ κα φ¨ + 3 φ˙ − m 2φ−(α+1) = 0, (4.1) a 2 p a˙ 2 8π = 2 (ρm + ρφ), a 3mp m 2 ρ = p φ˙2 + κm 2φ−α . φ 32π p

Here an over-dot denotes a derivative with respect to time, a is the scale factor, ρm is the energy density of nonrelativistic (cold dark and baryonic) matter, ρφ is that of the dark

−1/2 energy scalar ﬁeld φ, mp = G is the Planck mass where G is the gravitational constant, and α > 0 is a free parameter of the potential energy density of φ and determines κ which is [see7, 45]

8 α + 4 2 α/2 κ = α(α + 2) . (4.2) 3 α + 2 3

In the limit α 7−→ 0 the φCDM model reproduces the spatially-ﬂat ΛCDM cosmology while in the limit α 7−→ ∞ it reduces to the Einstein–De Sitter model with no DE but only CDM and baryonic matter. The value of α determines the rapidity of the time-evolution of the DE density, with a larger α corresponding to more rapidly decreasing DE density. The cosmological parameters of the φCDM model are p = (Ωm0, α). Many diﬀerent data sets have been used to derive constraints on the parameters of the three models we consider here.1 In this paper we use growth-factor measurements to

1See, e.g., [123], [49], [115], [124], [125], [126], [127], and [128]; also see [129]. For constraints on these

64 constrain cosmological parameters,2 under the assumption that GR is the correct model of gravity. Growth factor data have previously been used to test GR. Here we ﬁnd that if we assume GR then growth-factor measurements provide tight constraints on cosmological parameters. This chapter is organized as follows. In the next section we discuss the data and the analysis techniques that we use to derive cosmological parameter constraints. In Sec. 4.3, we present and discuss our results. Chapter5 contains our conclusions.

4.2 Data and analysis

We use three diﬀerent types of data to constrain cosmological parameters: the growth rate of large-scale structure (LSS) measurements; supernova type Ia (SNIa) distance-modulus measurements as a function of redshift; and Hubble parameter measurements.

4.2.1 Growth rate of LSS

In linear perturbation theory the nonrelativistic (cold dark and baryonic) matter density perturbation δm = δρm/ρm obeys

¨ a˙ ˙ 4π δm + 2 δm − 2 ρmδm = 0, (4.3) a mp

where the scale factor a, with current value a0, is related to redshift z through 1 + z = a0/a. The analytic growing solution of (4.3) is

Z ∞ 0 5Ωm0E(z) 1 + z 0 δm(t) ∝ D(z) = 3 0 dz , (4.4) 2 z E (z ) and related models from near-future data see [77], [130], [131], and references therein. 2For related work with growth factor data, see [132].

65 z A(z) σ Reference 0.067 0.4230 0.0550 1 0.150 0.3900 0.0800 2 0.170 0.5100 0.0600 3 0.220 0.4200 0.0700 4 0.250 0.3512 0.0583 5 0.350 0.4400 0.0500 3 0.370 0.4602 0.0378 5 0.410 0.4500 0.0400 4 0.550 0.5000 0.0700 6 0.570 0.4150 0.0340 7 0.600 0.4300 0.0400 4 0.770 0.4900 0.1800 8 0.780 0.3800 0.0400 4 0.800 0.4700 0.0800 9

Table 4.1: Growth parameter measurements and 1σ uncertainties. Reference number shown in the last column: 1. [133], 2. [134], 3. [135], 4. [64], 5. [65], 6. [136], 7. [66], 8. [63], 9. [137].

where E(z) = H(z)/H0 and H(z) is the Hubble parameter whose current value is the Hubble constant H0. D(z) is normalized such that D(z = 0) = 1. Note, that the analytic solution (4.4) is valid only for, in general spatially non-ﬂat, ΛCDM cosmology. In cosmological models where dark energy density is allowed to evolve in time, Eq. (4.3) has to be solved numerically, which we do, in order to compute growth factor D(z) for XCDM and φCDM cosmological models. The observable we use in our analysis is constructed from the linear theory, redshift-

−1 −1 dependent rms mass ﬂuctuations in 8h Mpc spheres (where h is H0 in units of 100 km s

−1 0 0 Mpc ), σ8(z) = σ8D(z), where σ8 is the current value of σ8(z). We shall also need f(z), the logarithmic derivative of the matter density perturbation D(z) with respect to the scale factor a, f(z) = d ln D/d ln a. Using (4.4) we ﬁnd an analytic expression for f(z) that we use to compute growth factor in the ΛCDM cosmological model

aa¨ 5Ω (1 + z)2 f(z) = − 1 + m0 . (4.5) a˙ 2 2 E2(z)D(z)

66 For XCDM and φCDM cosmological models we compute f(z) numerically. The observable we use is the growth parameter Aobs(z) = f(z)σ8(z) that also accounts for the Alcock-

0 Paczynski eﬀect in redshift-space distortions. The model prediction at redshift z is Ath(z, σ8, p) = 0 f(z, p)σ8(z, σ8, p) where p is the vector of cosmological parameters. We use a χ2 analysis to derive constraints on cosmological parameters from growth factor

2 0 data. χ depends on the cosmological parameters p and σ8,

N 0 2 X [Ath(zi, σ , p) − Aobs(zi)] χ2 (σ0, p) = 8 . (4.6) G 8 σ2 i=1 i

Here N is the number of data points and σi is the 1σ uncertainty on measurement Aobs(zi) at

3 0 redshift zi, see Table 4.1. For our purposes, σ8 is a nuisance parameter that we marginalize 0 over. To do so we assume a Gaussian prior for σ8 determined from cluster observations by 0 −0.47 [138], for spatially-ﬂat ΛCDM, with mean σ8(Ωm0) = 0.813(Ωm0/0.25) and 1σ uncer- 2 2 1/2 −0.47 tainty σ 0 (Ωm0) = (σ 0 +b ) (Ωm0/0.25) , where the statistical uncertainty σσ0 = 0.012 σ8 σ8 8 and the systematic uncertainty b = 0.02 are added in quadrature. [138] note that this relation is also adequate in the non-ﬂat ΛCDM model and for alternative background cosmologies.4 Then the posterior probability density function that depends only on the cosmological parameters p is given by

∞  2  h 0 0 i 1 R  χ2 (σ0, p) σ8 − σ8(Ωm0)  L (p) = √ exp − G 8 − dσ0. (4.7) G 2 2σ2 (Ω ) 8 σ 0 (Ωm0) 2π  0 m0  σ8 0  σ8 

2 Finally, we compute the marginalized χG(p) = −2 ln(LG(p)), and minimize this with respect to parameters p to find the best-fit parameter values p0. We also compute 1σ, 2σ, and 3σ 3For the redshift z = 0.57 bin we use the value for model 2 from Table 1 of [66] and an average of the upper and lower 1σ uncertainties given for that model. 4In this preliminary analysis we use this approximate, empirical expression for illustrative purposes. 0 However, σ8 does (weakly) depend on the full set of cosmological parameters p in its own way for every cosmological model, so our analyses are approximate. Given that our results, described below, are encouraging, a more careful analysis that accounts for this effect is warranted and will be discussed elsewhere.

67 2 2 2 cosmological parameter conﬁdence contours bounded by χG(p) = χG(p0) + 2.3, χG(p) = 2 2 2 χG(p0) + 6.17, and χG(p) = χG(p0) + 11.8, respectively.

4.2.2 SNIa distance modulus

The largest set of data we use are the 580 Type Ia supernova distance modulus µobs(z) measurements from the [12] Union 2.1 compilation (covering the redshift range of 0.015 ≤ z ≤ 1.414). The predicted distance-modulus is

µth(z) = 5 log10[3000y(z)(1 + z)] + 25 − 5 log10(h), (4.8) where y(z) is the dimensionless coordinate distance

 z  1 Z dz0 √ p y(z) = sin  −Ωk 0  , (4.9) −Ωk E(z ) 0

and Ωk is the spatial curvature density parameter. Since the SNIa distance modulus mea-

2 surements µobs are correlated we use χ deﬁned through the inverse covariance matrix

2 T −1 χSN (h, p) = ∆µ C ∆µ. Here the vector of diﬀerences ∆µi = µth(zi,H0, p) − µobs(zi), and C−1 is the inverse of the 580 by 580 Union 2.1 compilation covariance matrix.

4.2.3 Hubble parameter

We use 20 Hubble parameter measurements Hobs(z) and 1σ uncertainties covering redshift range 0.09 ≤ z ≤ 2.3 [113, 139–141], as listed in Table 1 of [114]. We only include independent measurements of the Hubble parameter, i.e., we exclude Hobs(z) points that are possibly correlated with growth factor measurements in Table 4.1 above. Theoretical expressions for the Hubble parameter follow directly from the Friedmann

68 equation in each model. In the case of the ΛCDM model,

2 2 3 2 Hth(z, p) = H0 Ωm0(1 + z) + (1 − Ωm0 − ΩΛ)(1 + z) + ΩΛ , (4.10) while for the spatially-ﬂat XCDM parameterization,

2 2 3 3(1+wX ) Hth(z, p) = H0 Ωm0(1 + z) + (1 − Ωm0)(1 + z) , (4.11) and for the spatially-ﬂat φCDM model,

1 H2 (z, p) = H2Ω (1 + z)3 + φ˙2 + κm 2φ−α . (4.12) th 0 m0 12 p

We use the same technique to constrain cosmological parameters from H(z) measure-

2 ments as we used in Sec. 4.2.1 for the growth factor data analysis. First, we deﬁne χH (H0, p) in accordance with Eq. (4.6) where instead of the growth factor A(z) we insert the Hubble parameter H(z).

4.2.4 Computation of joint χ2(p)

We perform two joint analyses, one for the combination of SNIa and H(z) data, the other for all three data sets. For the SNIa+H(z) analysis we multiply likelihood functions from the SNIa data and the H(z) data and then marginalize this over the nuisance parameter

−1 −1 H0 with a Gaussian prior with mean value H0 = 68.0 km s Mpc and 1σ uncertainty

−1 −1 σH0 = 2.8 km s Mpc ([117], also see [142], [143], [144]) to finally determine the joint 2 χSNIa+H (p) function, which depends only on cosmological parameters p. This is then used to find the best-fit values of p0 and corresponding cosmological parameter constraints. The second joint analysis, of the SNIa+H(z) data with the growth factor data, is based on

2 2 2 2 adding their χ -functions, χJnt(p) = χSNIa+H (p) + χG(p).

69 4.3 Results and discussion

We derived cosmological parameter constraints from combination of SNIa+H(z) data sets as well as from a joint analysis of all three data sets.5 Our results, presented in the form of isocontours in cosmological parameter space, are shown in Figs. 4.1, 4.2, and 4.3 for the ΛCDM, XCDM and φCDM models, respectively. In the ΛCDM model the growth factor data favor higher best-fit value of a negative spatial curvature parameter Ωk0 = 1 − Ωm0 − ΩΛ (which corresponds to a closed, spherical spatial geometry) along with a higher best-fit value of Ωm0 compared to what other cosmological tests favor, such as SNIa, Hubble parameter measurements, BAO and CMB (see for example [123]-[129] and references therein). In the case of the XCDM parameterization the growth factor data favor a steeper time dependence of dark energy density and also a higher value of ordinary matter energy density parameter (i.e. equation of state parameter wX has a lower best-fit value and Ωm0 has a higher best-fit value) in comparison with constraints derived from the above-mentioned data sets. However, for the φCDM model one observes consistent results for the best-fit values of cosmological parameters (Ωm0, α) with those previously obtained using the data sets mentioned above. Also, our results for the ΛCDM model differ from constraints obtained for this model from other analyses of growth factor data (see [133], [66]). We suspect that the reason for this and the reason that the constraining power of growth rate data has not previously been recognized is because these data have almost always been used to constrain cosmological parameters in the context of modified gravity models. These modified gravity models have more free parameters than the models we have considered here, because we have assumed that general relativity provides an adequate description of gravitation on cosmological scales. The other striking feature of the growth rate data constraints is that for all three models they align well with those of the SNIa+H(z) joint constraints.

5We have not used all the H(z) measurements in this paper, excluding points from Table 1 of [114] that are possibly correlated with some of the growth rate data we use in this paper.

70 No Big Bang 1.2

1 Closed Open

0.8

Ω 0.6

0.4

Acceleration 0.2 Deceleration

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Ω m

Figure 4.1: 1, 2, and 3σ constraint contours for the ΛCDM model from: growth factor measurements [blue dashed lines with blue filled circle at best-fit (Ωm, ΩΛ) = (0.41, 0.87) with 2 χmin/dof = 7.65/12]; SNIa+H(z) apparent magnitude data [red dot-dashed lines with red 2 filled circle at best-fit (Ωm, ΩΛ) = (0.23, 0.59) with χmin/dof = 562/598]; and a combination of all data sets [black solid lines and black filled circle at best-fit (Ωm, ΩΛ) = (0.28, 0.69) 2 with χmin/dof = 571/612]. The dashed straight line corresponds to spatially-flat models, the dotted line demarcates zero acceleration models, and the area in the upper left-hand corner is the region for which there is no big bang.

71 −0.2

−0.4

−0.6 Deceleration

Acceleration −0.8

−1 Flat ΛCDM X ω −1.2

−1.4

−1.6

−1.8

−2 0 0.1 0.2 0.3 0.4 0.5 0.6 Ω m

Figure 4.2: 1, 2, and 3σ constraint contours for the XCDM model from: growth factor measurements [blue dashed lines with blue filled circle at best-fit (Ωm, wX ) = (0.36, -1.34) 2 with χmin/dof = 7.70/12]; SNIa+H(z) apparent magnitude data [red dot-dashed lines with 2 red filled circle at best-fit (Ωm, wX ) = (0.27, -0.90) with χmin/dof = 562/598]; and a combination of all data sets [black solid lines and black filled circle at best-fit (Ωm, wX ) = (0.28, 2 -0.94) with χmin/dof = 571/612]. The dashed straight line corresponds to spatially-flat ΛCDM models and the dotted curved line demarcates zero acceleration models.

72 5

4.5

3.5

α 2.5

1.5

Deceleration 1 Acceleration

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Ω m

Figure 4.3: 1, 2, and 3σ constraint contours for the φCDM model from: growth factor measurements [blue dashed lines with blue filled circle at best-fit (Ωm, α) = (0.28, 0.052) with 2 χmin/dof = 8.62/12]; SNIa+H(z) apparent magnitude data [red dot-dashed lines with red 2 filled circle at best-fit (Ωm, α) = (0.26, 0.302) with χmin/dof = 562/598]; and a combination of all data sets [black solid lines and black filled circle at best-fit (Ωm, α) = (0.27, 0.300) 2 with χmin/dof = 570/612]. The dotted curved line demarcates zero acceleration models and the horizontal α = 0 axis corresponds to spatially-flat ΛCDM models.

73 Chapter 5

Conclusion

Chapter2 presents a forecast of the precision at which planned near-future space-based spec- troscopic galaxy surveys should be able to constrain the time dependence of dark energy density. For the first time, was used a consistent physical model of time-evolving dark energy, φCDM, in which a minimally-coupled scalar field slowly rolls down its self-interaction potential energy density. It has been shown that if general relativity is assumed, the deviation of the parameter α of the φCDM model can be constrained to better than 0.05; this is almost an order of magnitude better than the best currently available result. The constraints on basic cosmological parameters, such as the relative energy densities of non-relativistic matter and spatial curvature, depend on the adopted dark energy model. However, in the φCDM model the expected constraints are more restrictive than those derived using the XCDM or wCDM parameterizations. This is due to the fact that the φCDM model has fewer parameters. Also, the XCDM and wCDM parameterizations assign equal weight to all possible values of w, while in the φCDM model there is an implicit theoretical prior on which equation of state parameter values are more likely, based on how easy it is to produce such a value within the model. Since the observational consequences of dark energy and modified gravity are partially degenerate, constraints on modified gravity parameters will depend on the assumptions

74 made about dark energy. The constraints on γ are most restrictive in the ΛCDM model. For the φCDM model the constraints on γ are about a third tighter than those for the ωCDM and XCDM parameterizations. These results are very encouraging: data from an experiment of the type we have modeled will be able to provide very good, and probably revolutionary, constraints on the time evolution of dark energy. In Chapter3 the original φCDM model was extended to nonflat geometries and it was shown that in the curvature-dominated epoch the solution is also an attractor or time- dependent fixed point (see Sec. 3.2.1 and the AppendixB). Predictions of the model for an illustrative set of cosmological tests were computed and shown that the presence of space curvature will broaden the allowed range of model parameters. Spatial curvature should be considered as a free parameter when observational data are analyzed. The nonflat φCDM model we have developed here is the only consistent nonflat time-variable DE model to date and can be used as a fiducial model for such analyses. The computations have shown that for a single cosmological test there is a degeneracy point in parameter space for each fixed value of the cosmological observable of the test. At this point one cannot differentiate between contours corresponding to different values of spatial curvature. However, these points of degeneracy do not coincide in (Ωm0, α) parameter space for the different cosmological tests. Hence it is important to use multiple cosmological tests in order to determine spatial curvature from observations. A joint analysis of geometry and growth factor measurements appears to be a fruitful way to constrain space curvature. CMB anisotropy data will also likely provide useful constraints on space curvature. This will first require accounting for spatial curvature effects on the quantum-mechanical zero-point fluctuations generated during inflation, which will affect the primordial density perturbations power spectrum [145–153]. While the curved- space computation is more involved than the corresponding flat-space one, the resulting constraints from CMB anisotropy data on space curvature in the presence of dynamical

75 dark energy are likely to prove quite interesting. In Chapter4 three general-relativistic DE cosmological models were used to analyze the largest collection of growth factor measurements to date. It was shown that growth factor data constraints on cosmological parameters are quite restrictive, roughly close to those from joint SNIa+H(z) and baryon acoustic oscillations (BAO) peak length-scale measurements, and less restrictive than those from cosmic microwave background (CMB) anisotropy observations. These growth factor results must be viewed as tentative, given that this is an area of research that is still under active development. It is important to continue to study possible sources of systematic uncertainty – and given the diﬀerences we have found between growth rate data constraints and these from SNIa and H(z) measurements, it is not unreasonable to suspect that there might be an as yet hidden source of systematic uncertainty. It is, however, clear that growth factor measurements will soon be able to provide cosmological constraints as restrictive and as reliable as those from CMB anisotropy, BAO, H(z), and SNIa measurements.

76 Bibliography

[1] P. A. R. Ade, et al., arXiv:1303.5076 (2013).

[2] S. Dodelson, MODERN COSMOLOGY, ACADEMIC PRESS, 2003.

[3] S. Capozziello and M. D. Laurentis, arXiv:1307.4523 (2013).

[4] M. Sami and R. Myrzakulov, arXiv:1309.4188 (2013).

[5] J. Martin, Comptes Rendus Physique 13, 566 (2012).

[6] P. J. E. Peebles, Astrophys. J. 284, 439 (1984).

[7] P. J. E. Peebles and B. Ratra, Astrophys. J. Lett. 325, L17 (1988).

[8] S. Tsujikawa, Class. Quant. Grav. 30, 214003 (2013).

[9] J. Sol`a,J. Phys. Conf. Ser. 453, 012015 (2013).

[10] R. Jimenez, arXiv:1307.2452 (2013).

[11] C. P. Burgess, arXiv:1309.4133 (2013).

[12] N. Suzuki, et al., Astrophys. J. 746, 85 (2012).

[13] A. Conley, et al., Astrophys. J. Suppl. 192, 1 (2011).

[14] X.-D. Li, et al., JCAP 1107, 011 (2011).

[15] A. Barreira and P. P. Avelino, Phys. Rev. D84, 083521 (2011).

[16] B. Ratra and M. S. Vogeley, Publ. Astron. Soc. Pac. 120, 235 (2008).

77 [17] E. Komatsu, et al., Astrophys. J. Suppl. 192, 18 (2011).

[18] C. L. Reichardt, et al., Astrophys. J. 749, L9 (2012).

[19] G. Chen and B. Ratra, Publ. Astron. Soc. Pac. 115, 1143 (2003).

[20] W. J. Percival, et al., Mon. Not. Roy. Astron. Soc. 401, 2148 (2010).

[21] M. A. Dantas, et al., Phys. Lett. B699, 239 (2011).

[22] A. Carnero, et al., Mon. Not. Roy. Astron. Soc. 419, 1689 (2012).

[23] L. Anderson, et al., arXiv:1203.6594 (2012).

[24] K.-H. Chae, et al., Astrophys. J. 607, L71 (2004).

[25] S. Lee and K.-W. Ng, Phys. Rev. D76, 043518 (2007).

[26] M. Biesiada, et al., Mon. Not. Roy. Astron. Soc. 406, 1055 (2010).

[27] L. Samushia and B. Ratra, Astrophys. J. 650, L5 (2006).

[28] A. A. Sen and R. J. Scherrer, Phys. Lett. B659, 457 (2008).

[29] N. Pan, et al., Class. Quantum Grav. 27, 155015 (2010).

[30] Y. Chen and B. Ratra, Phys. Lett. B703, 406 (2011).

[31] M. Baldi and V. Pettrino, Mon. Not. Roy. Astron. Soc. 412, L1 (2011).

[32] C. D. Boni, et al., Mon. Not. Roy. Astron. Soc. 415, 2758 (2011).

[33] N. Brouzakis, et al., JCAP 03, 049 (2011).

[34] L. Campanelli, et al., arXiv:1110.2310 (2011).

[35] S. W. Allen, et al., Mon. Not. Roy. Astron. Soc. 383, 879 (2008).

78 [36] L. Samushia and B. Ratra, Astrophys. J. 680, L1 (2008).

[37] M. Tong and H. Noh, Eur. Phys. J. C71, 1586 (2011).

[38] A. Blanchard, A&AR 18, 595 (2010).

[39] D. Sapone, Int. J. Mod. Phys. A25, 5253 (2010).

[40] R. Jimenez, Fortschr. Phys. 59, 602 (2011).

[41] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003).

[42] L. Perivolaropoulos, J. Phys. Conf. Ser. 222, 012024 (2010).

[43] B. Ratra, Phys. Rev. D43, 3802 (1991).

[44] S. Podariu and B. Ratra, Astrophys. J. 532, 109 (2000).

[45] B. Ratra and P. J. E. Peebles, Phys. Rev. D37, 3406 (1988).

[46] G. Chen and B. Ratra, Astrophys. J. 612, L1 (2004).

[47] K. M. Wilson, et al., Mod. Phys. Lett. A21, 2197 (2006).

[48] Y. Chen and B. Ratra, A&A 543, A104 (2012).

[49] D. Mania and B. Ratra, Phys. Lett. B715, 9 (2012).

[50] L. Samushia, CONSTRAINING SCALAR FIELD DARK ENERGY WITH COSMO- LOGICAL OBSERVATIONS, PhD thesis, Kansas State University, 2009.

[51] R.-J. Yang, et al., Int. J. Mod. Phys. A26, 317 (2011).

[52] A. V. Frolov and J.-Q. Guo, arXiv:1101.4995 (2011).

[53] N. J. Nunes, et al., Phys. Rev. D83, 083523 (2011).

[54] J. Grande, et al., JCAP 1108, 007 (2011).

79 [55] R. Saitou and S. Nojiri, Eur. Phys. J. C71, 1712 (2011).

[56] R. Silva, et al., A&A 537, A11 (2012).

[57] A. Y. Kamenshchik, et al., Phys. Lett. B702, 191 (2011).

[58] M. Maggiore, et al., Phys. Lett. B704, 102 (2011).

[59] C. Blake, et al., Mon. Not. Roy. Astron. Soc. 418, 1707 (2011).

[60] F. Beutler, et al., Mon. Not. Roy. Astron. Soc. 416, 3017 (2011).

[61] W. J. Percival, et al., Mon. Not. Roy. Astron. Soc. 385, L78 (2004).

[62] R. E. Angulo, et al., Mon. Not. Roy. Astron. Soc. 387, 921 (2008).

[63] L. Guzzo, et al., Nature 451, 541 (2008).

[64] C. Blake, et al., Mon. Not. Roy. Astron. Soc. 415, 2876 (2011).

[65] L. Samushia, et al., Mon. Not. Roy. Astron. Soc. 420, 2102 (2012).

[66] B. A. Reid, et al., arXiv:1203.6641 (2012).

[67] R. Laureijs, et al., arXiv:1110.3193 (2011).

[68] J. Green, et al., arXiv:1108.1374 (2011).

[69] Y. Wang, et al., Mon. Not. Roy. Astron. Soc. 409, 737 (2010).

[70] L. Samushia, et al., Mon. Not. Roy. Astron. Soc. 410, 1993 (2010).

[71] E. Majerotto, et al., Mon. Not. Roy. Astron. Soc. 424, 1392 (2012).

[72] T. Basse, et al., arXiv:1205.0548 (2012).

[73] S. Podariu, et al., Astrophys. J. 553, 39 (2001).

80 [74] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010).

[75] S. Tsujikawa, Lectures on Cosmology Accelerated Expansion of the Universe by Georg Wolschin, volume 800, Berlin: Springer, 1999.

[76] S. Capazziello and M. D. Laurentis, Phys. Rept. 509, 167 (2011).

[77] A. Pavlov, et al., Astrophys. J. 760, 19 (2012).

[78] W. Fischler, Nucl. Phys. B259, 730 (1985).

[79] N. Kaiser, Mon. Not. Roy. Astron. Soc. 227, 1 (1987).

[80] E. V. Linder, Phys. Rev. D72, 043529 (2005).

[81] C. Alcock and B. Paczy´nski,Nature 281, 358 (1979).

[82] M. Chevallier and D. Polarski, Int. J. Mod. Phys. D10, 213 (2001).

[83] E. V. Linder, Phys. Rev. Lett. 90, 091301 (2003).

[84] A. Albrecht, et al., arXiv:0901.0721 (2009).

[85] L. Perotto, et al., JCAP 0610, 013 (2006).

[86] M. Martinelli, et al., Phys. Rev. D83, 023012 (2011).

[87] A. Orsi, et al., Mon. Not. Roy. Astron. Soc. 405, 1006 (2010).

[88] J. E. Geach, et al., Mon. Not. Roy. Astron. Soc. 402, 1330 (2010).

[89] H. K. Jassal, et al., Mon. Not. Roy. Astron. Soc. 405, 2639 (2010).

[90] T. M. Davis, et al., Astrophys. J. 666, 716 (2007).

[91] D. H. Weinberg, et al., arXiv:1306.0913 (2013).

[92] G. Hinshaw, et al., Astrophys. J. Suppl. 208, 19 (2013).

81 [93] S. Podariu, et al., Astrophys. J. 559, 9 (2001).

[94] R. Aurich and F. Steiner, Mon. Not. Roy. Astron. Soc. 334, 735 (2002).

[95] R. Aurich and F. Steiner, Int. J. Mod. Phys. 13, 123 (2004).

[96] R. Aurich and F. Steiner, Phys. Rev. D67, 123511 (2003).

[97] J. L. Crooks, Astropart. Phys. 20, 361 (2003).

[98] K. Ichikawa and T. Takahashi, Phys. Rev. D73, 083526 (2006).

[99] K. Ichikawa and T. Takahashi, JCAP 02, 001 (2007).

[100] K. Ichikawa and T. Takahashi, JCAP 04, 027 (2008).

[101] E. L. Wright, et al., arXiv:astro-ph/0603750 (2006).

[102] K. Ichikawa, et al., JCAP 12, 005 (2006).

[103] G.-B. Zhao, et al., Phys. Lett. B648, 8 (2007).

[104] Y. Wang and P. Mukherjee, Phys. Rev. D76, 103533 (2007).

[105] Y. Gong, et al., Astrophys. J. 681, 27 (2008).

[106] J.-M. Virey, et al., JCAP 12, 008 (2008).

[107] Y. Wang and P. Mukherjee, Phys. Rev. D80, 123504 (2009).

[108] J. Martin, Mod. Phys. Lett. A23, 1252 (2008).

[109] K. Thepsuriya and B. Gumjudpai, arXiv:0904.2743 (2009).

[110] Z.-Q. Chen and D.-H. Guo, Int. J. Theor. Phys. 51, 3856 (2012).

[111] B. Gumjudpai and K. Thepsuriya, Astrophys. Space Sci. 342, 537 (2012).

82 [112] A. Pavlov, et al., Phys. Rev. D88, 123513 (2013).

[113] N. G. Busca, et al., Astron. Astrophys. 552, A96 (2013).

[114] O. Farooq and B. Ratra, Astrophys. J. Lett. 766, L7 (2013).

[115] O. Farooq, et al., Phys. Lett. B726, 72 (2013).

[116] O. Farooq, et al., Astrophys.Space Sci. 357, 11 (2015).

[117] G. Chen and B. Ratra, Publ. Astron. Soc. Pac. 123, 1127 (2011).

[118] J. A. Newman and M. Davis, Astrophys. J. Lett. 534, L11 (2000).

[119] D. Huterer and M. S. Turner, Phys. Rev. D64, 123527 (2001).

[120] S. Podariu and B. Ratra, Astrophys. J. 563, 28 (2001).

[121] B. Ratra, Phys. Rev. D45, 1913 (1992).

[122] A. Pavlov, et al., Phys. Rev. D90, 023006 (2014).

[123] L. Samushia, et al., arXiv:0706.1963 (2007).

[124] A. Piloyan, et al., JCAP 1307, 042 (2013).

[125] O. Akarsu, et al., Eur. Phys. J. Plus 129, 22 (2014).

[126] M. Li, et al., JCAP 1309, 021 (2013).

[127] E. L. D. Perico, et al., Phys. Rev. D88, 063531 (2013).

[128] V. H. C´ardenas,et al., arXiv:1306.0779 (2013).

[129] S. Crandall and B. Ratra, arXiv:1311.0840 (2013).

[130] S. A. Appleby and E. V. Linder, Phys. Rev. D87, 023532 (2013).

83 [131] M. Arabsalmani, et al., Phys. Rev. D87, 083001 (2013).

[132] N. A. Arkhipova, et al., arXiv:1406.0407 (2014).

[133] F. Beutler, et al., Mon. Not. Roy. Astron. Soc. 423, 3430 (2012).

[134] C. Hawkins, et al., Mon. Not. Roy. Astron. Soc. 346, 78 (2003).

[135] Y.-S. Song and W. J. Percival, JCAP 0910, 004 (2009).

[136] N. P. Ross, et al., Mon. Not. Roy. Astron. Soc. 381, 573 (2007).

[137] A. Torre, et al., arXiv:1303.2622 (2013).

[138] A. Vikhlinin, et al., Astrophys. J. 692, 1060 (2009).

[139] L. V. J. Simon and R. Jimenez, Phys. Rev. D71, 123001 (2005).

[140] D. Stern, et al., JCAP 1002, 008 (2010).

[141] M. Moresco, et al., JCAP 1208, 006 (2012).

[142] J. R. Gott, et al., JCAP 549, 1 (2001).

[143] G. Chen, et al., Publ. Astron. Soc. Pac. 115, 1269 (2003).

[144] E. Calabrese, et al., Phys. Rev. D86, 043520 (2012).

[145] J. R. Gott, Nature 295, 304 (1982).

[146] B. Ratra and P. J. E. Peebles, Astrophys. J. Lett. 432, L5 (1994).

[147] B. Ratra and P. J. E. Peebles, Phys. Rev. D52, 1837 (1995).

[148] M. Kamionkowski, et al., Astrophys. J. 434, L1 (1994).

[149] M. Bucher, et al., Phys. Rev. D52, 3314 (1995).

84 [150] D. H. Lyth and A. Woszczyna, Phys. Rev. D52, 3338 (1995).

[151] K. Yamamoto, et al., Astrophys. J. 455, 412 (1995).

[152] K. Ganga, et al., Astrophys. J. 484, 517 (1997).

[153] M. G´orski,et al., Astrophys. J. Suppl. 114, 1 (1998).

[154] M. Tegmark, Phys. Rev. Lett. 79, 3806 (1997).

[155] D. J. Eisenstein and W. Hu, Astrophys. J. 496, 605 (1998).

[156] B. Ratra, Phys. Rev. D50, 5252 (1994).

85 Appendix A

Appendix for Chapter 2

In this Appendix we summarize how to estimate the precision of measurements from the survey parameters. The Fisher matrix coeﬃcients are given by

1 Z kmax ∂ ln P ∂ ln P d3k Fij = i j Veff (k, µ) 3 , (A.1) 2 kmin ∂p ∂p (2π) where the effective volume is nP (k, µ) V = V , (A.2) eff 0 1 + nP (k, µ) and V0 is the total survey volume and n is the number density. Also, following [154], we

2 dr(z) multiply the integrand in Eq. (A.1) by a Gaussian factor exp −k σz dz , where r(z) is the co-moving distance, in order to account for the errors in distance induced by the errors of redshift measurements, σz = 0.001. We model the theoretical power spectrum using an analytic approximation of [155]. We integrate in k from kmin = 0 to kmax, where the kmax values depend on redshift and are chosen in such a way that the small scales that are dominated by non-linear effects are excluded. The range of scales that will be fitted to the future surveys will depend on how well the theoretical templates are able to describe small-scale clustering and is difficult to predict. The kmax values along with the expected

86 bias and number density of galaxies are listed in Table A.1. In order to derive the Fisher matrix of a speciﬁc cosmological model we have to go from our initial parameter space to the parameter space of the cosmological model whose Fisher matrix we want. The transformation formula for the Fisher matrix is given by [see, e.g., 84, for a review]

0 ∂pi ∂pj Flm = 0 0 Fij, (A.3) ∂pl ∂pm where the primes denote the “new” Fisher matrix and parameters. We now list the derivatives of the transformation coefficients of the φCDM model in the limit α −→ 0 and Ωk −→ 0 (which corresponds to the fiducial spatially-flat ΛCDM model).

The transformation coeﬃcients relating fk(z) and the parameters (h, Ωm, Ωk, α) are

∂f (z) 1 k = − , (A.4) ∂h h

∂fk(z) 1 3 = 2 [1 − (1 + z) ], (A.5) ∂Ωm 2E(z)

∂fk(z) 1 2 = 2 [1 − (1 + z) ], (A.6) ∂Ωk 2E(z) ∂f (z) (1 − Ω ) k = − m . (A.7) ∂α 8E(z)2

For the other transformation coeﬃcients, it is convenient to introduce the integral

Z z dz0 D(z) = 0 . (A.8) 0 E(z )

Then the transformation coeﬃcients between f⊥(z) and the parameters (h, Ωm, Ωk, α) are

∂f (z) 1 ⊥ = − , (A.9) ∂h h

Z z 0 ∂f⊥(z) 1 dz 0 3 = 0 3 [1 − (1 + z ) ], (A.10) ∂Ωm 2D(z) 0 E(z )

87 2 Z z 0 ∂f⊥(z) D(z) 1 dz 0 2 = + 0 3 [1 − (1 + z ) ], (A.11) ∂Ωk 6 2D(z) 0 E(z ) Z z 0 ∂f⊥(z) (1 − Ωm) dz = − 0 3 . (A.12) ∂α 8D(z) 0 E(z ) Finally, the transformation coeﬃcients between the growth factor f(z) and the parameters (γ, h, Ωm, Ωk, α) are ∂f(z) f(z) = ln f(z), (A.13) ∂γ γ

∂f(z) γf(z) 2 3 = 2 E(z) − Ωm[(1 + z) − 1] , (A.14) ∂Ωm ΩmE(z)

∂f(z) γf(z) 2 = − 2 [(1 + z) − 1], (A.15) ∂Ωk E(z) ∂f(z) γf(z) = − [1 − Ω ]. (A.16) ∂α 4E(z)2 m

88 Table A.1: Values of the kmax, bias b(z) from [87], and the number densities n(z) taken from [88]. z kmax b(z) n(z) 0.55 0.144 1.0423 3220 0.65 0.153 1.0668 3821 0.75 0.163 1.1084 4364 0.85 0.174 1.1145 4835 0.95 0.185 1.1107 5255 1.05 0.197 1.1652 5631 1.15 0.2 1.2262 5972 1.25 0.2 1.2769 6290 1.35 0.2 1.2960 6054 1.45 0.2 1.3159 4985 1.55 0.2 1.4416 4119 1.65 0.2 1.4915 3343 1.75 0.2 1.4873 2666 1.85 0.2 1.5332 2090 1.95 0.2 1.5705 1613 2.05 0.2 1.6277 1224

89 Appendix B

Appendix for Chapter 3

In Sec. 3.2.1 we showed that the special time-dependent fixed point solution for the scalar field in the curvature-dominated epoch is stable if we ignore spatial inhomogeneities in the gravitational field. In this Appendix we show that gravitational spatial inhomogeneities do not spoil this property of the solution, thus preserving the inclination of the scalar field DE density to always want to try to dominate over the dominant energy density source [7, 45]. Inhomogeneities in the scalar field will induce inhomogeneities in the metric, and vice versa. We show that, in the curvature-dominated epoch, any slight inhomogeneities will die out. (This generalizes the flat-space results of Sec. IX of Ref. [45].) We linearize the disturbances in the metric about a curved Friedmann background metric in synchronous gauge. To this end, we write the line element as

2 µ ν µ ν ds =g ˜µνdx dx = (gµν + δgµν)dx dx . (B.1)

We work in time-orthogonal coordinates (t, r, θ, ϕ) with gµν given in Eq. (3.5) and the

90 perturbations

  0 0 0 0      0 f(r)hrr hrθ hrϕ  2   δgµν = a(t)   ,  0 h r2h h   rθ θθ θϕ    2 2 0 hrϕ hθϕ r sin (θ)hϕϕ (B.2)

2 where f(r) = 1/(1 − kr ), |hij| 1, and each hij is a function of t, r, θ, and ϕ. The scalar ﬁeld equation of motion in a space-time with cometricg ˜µν reads

˜ µν 0 ∇µ(˜g ∂νΦ) + V (Φ) = 0. (B.3)

The perturbed scalar ﬁeld is written

µ µ Φ(x ) = Φ0(t) + φ(x ) (B.4)

where φ is a small perturbation, |φ| |Φ0|, and Φ0 is a solution to the scalar ﬁeld equation of motion in the unperturbed homogeneous Friedmann background,

a˙ καm2 Φ¨ + 3 Φ˙ − p Φ−(α+1) = 0. (B.5) 0 a 0 2 0

Plugging (B.4) into (B.5) gives, to ﬁrst order in φ,

3˙a 1 1 φ¨ + φ˙ − ∇2φ + V 00(Φ )φ − h˙ Φ˙ = 0, (B.6) a a2 0 2 0

µν 2 where h = hrr + hθθ + hϕϕ = −g δgµν, and ∇ is the Laplacian for the three-dimensional

91 spacelike hypersurface of constant t in the unperturbed Friedman geometry,

1 ∂ ∂ ∂ ∇2 = (r2 − kr4) + kr (B.7) r2 ∂r ∂r ∂r 1 ∂ ∂ 1 ∂2 + sin(θ) + . r2 sin(θ) ∂θ ∂θ r2 sin2(θ) ∂ϕ2

When k = 0, ∇2 is the usual three-dimensional ﬂat-space Laplacian in spherical coordinates.

The tt component of the stress-energy tensor Qµν for Φ = Φ0 + φ, to ﬁrst order in φ, is

m2 h i m2 h i Q = p Φ˙ 2 + 2V (Φ ) + p Φ˙ φ˙ + V 0(Φ )φ , (B.8) tt 32π 0 0 16π 0 0

µν and the trace Q =g ˜ Qµν is, to ﬁrst order,

m2 h i m2 h i Q = p 4V (Φ ) − Φ˙ 2 + p 2φV 0(Φ ) − Φ˙ φ˙ . (B.9) 16π 0 0 8π 0 0

As for the Ricci tensor Rµν, we will also only require the tt component. To ﬁrst order it is

3¨a a˙ 1 R = − + h˙ + h¨ . (B.10) tt a a 2

By the Einstein ﬁeld equations (3.3) we therefore get the ﬁrst-order perturbation equation,

2˙a h¨ + h˙ = 2Φ˙ φ˙ − V 0(Φ )φ. (B.11) a 0 0

This corresponds to Eq. (3.14) of Ref. [147].

We now take a = a0t for the curvature-dominated epoch, where a0 is a constant of integration and we consider times t > 0. Thus, the system we need to analyze is

2 ¨ 3 ˙ L 00 1 ˙ ˙ φ + φ − 2 2 φ + V (Φ0)φ = hΦ0, (B.12) t a0t 2

92 2 h¨ + h˙ = 2Φ˙ φ˙ − V 0(Φ )φ. (B.13) t 0 0

Here we have made a hyperspherical harmonic transformation, the variables φ and h are now harmonic mode amplitudes, and L2 is the eigenvalue of the Laplacian operator (B.8) (see Ref. [147] and Sec. II of Ref. [156]). One has L2 → −1 (respectively L2 → −∞) in the limit of long wavelength (short wavelength) modes for the negative curvature case.

The ﬁeld Φ0 is the special solution obtained in Sec. 3.2.1, Eq. (3.9). We here write it as

m Φ0 = At , (B.14) where

2 m = , (B.15) α + 2 and A is, by (3.10) with n = 1,

καm2(α + 2)1/(α+2) A = p . (B.16) 8α + 24

Deﬁning

κα B = m2, (B.17) 2 p

Eqs. (B.12) and (B.13) can be rewritten as

3 J mA φ¨ + φ˙ + φ = ht˙ m−1, (B.18) t t2 2

2 h¨ + h˙ = 2mAtm−1φ˙ + BA−(α+1)tm−2φ, (B.19) t

93 2 2 2 2 where J = (α + 1)(m + 2m) − L /a0. As mentioned previously, L → −1 in the case that we are presently interested in (long-wavelength perturbations and negative curvature), so the constant J is a positive real number > 3.

−2 For the curvature-dominated case ρk ∝ t and so

ρ C2 Φ = t2m, (B.20) ρk where C is a constant of integration. Thus, Eqs. (B.18) and (B.19) can be written as

˙ r ¨ 3 ˙ J mAC h ρΦ φ + φ + 2 φ = , (B.21) t t 2 t ρk

r −(α+1) r ¨ 2 ˙ 2mBC ρΦ ˙ BA C ρΦ h + h = φ + 2 φ. t t ρk t ρk (B.22)

Following Ref. [45] we solve these equations by using a linear perturbation technique. Since p we are in the curvature-dominated epoch and ρΦ/ρk is small, we begin by searching for approximate solutions to (B.21) and (B.22) where the source terms on the right-hand side are neglected. That is, we ﬁrst solve the homogeneous equations (to get zeroth order solutions for φ and h˙ ),

3 J φ¨ + φ˙ + φ = 0, (B.23) 0 t 0 t2 0

2 h¨ + h˙ = 0, (B.24) 0 t 0 where subscript 0 now denotes solutions in zeroth order of the perturbation approach. Once we have these zeroth order solutions, we will plug them into the right-hand side of Eqs.

94 (B.18) and (B.19) in order to obtain new diﬀerential equations which can then be used to p p derive correction terms of order ρΦ/ρk. If our solutions with order ρΦ/ρk corrections are still decaying then it means that the stability result is established at least in the ﬁrst-order perturbation analysis. The zeroth order solution to (B.23) is

C h√ i C h√ i φ (t) = 1 cos J − 1 ln(t) + 2 sin J − 1 ln(t) , (B.25) 0 t t

where C1 and C2 are constants of integration, and the zeroth order solution to (B.24) is

C h (t) = 3 + C , (B.26) 0 t 4

where C3 and C4 are constants of integration. Note that, up to oscillatory bounded functions

−(α+4)/(α+2) −2 −1 of time, φ0/Φ0 ∝ t ∈ (t , t ), so we confirm the result of Sec. 3.2.1 that if we ignore the effect of metric perturbations the time-dependent fixed-point solution is stable.

Writing φ = φ0 + φ1 and h = h0 + h1, and plugging (B.26) into Eq. (B.18), we get for the ﬁrst order φ1 equation

3 J mA φ¨ + φ˙ + φ = h˙ tm−1. (B.27) 1 t 1 t2 1 2 0

(We shall not need the h1 differential equation.) Solving this differential equation for φ1 we find

mAC tm−1 φ (t) = − 3 . (B.28) 1 2(m2 − 1 + J)

From this solution and that in (B.25), we ﬁnd, up to oscillatory bounded functions of time, p φ1(t)/φ0(t) ∝ ρΦ/ρk, so in the curvature-dominated epoch, where ρΦ ρk, the correction to the scalar ﬁeld solution from the metric inhomogeneity is small.